I spent one really miserable night because I found out during the rainstorm that my bivy sack wasn't waterproof. Drying it out wasn't an option because of the rain and moving some where else also wasn't an option because of the terrain.
My temporary solution was to sleep in my rain gear (pants and jacket) inside my sleeping bag, is there a better solution for a situation like this?
First - do everything to prevent getting a wet bag!
A wet sleeping bag must be a nightmare - the best solution is to make sure this never happens to you again!
Before I use a new shelter somewhere remote I would wait for bad weather and test it out in an exposed location where I can retreat easily if things go pear-shaped. I always inspect my shelter carefully before heading for the wilds to ensure that the seams and fabric are in good condition. And I'd always take a shelter that had a bit in hand to cope with the worst conditions I'm likely to encounter. Whether by luck or by judgement, I've never had a wet bag in decades of wild camping.
Carry sufficient dry clothing
Of course there is always the risk of shelter failure. So I would always carry enough clothing to ensure I could survive if my bag got soaked. This means at least a base-layer, puffy, sleeping socks and hat that I keep dry during the day. With a wet bag I'd wear these under my walking clothing and rainwear. I wouldn't be comfortable, but at least I'd make it out.
With a wet bag, consider getting back on the trail
I always carry a decent walking torch and lightweight spare, and in almost all circumstances I'd probably prefer to get on the move if my bag was unusable - I'm going to be warmer and psychologically I'd find it less miserable to be doing something rather than shivering in the dark.
Otherwise, you're in survival mode!
If hitting the trail was impossible for some reason, you're in survival mode.
I'm going to assume that you're too deep into the bush to bail and get back to civilization, and that your bag is all that's standing between you and hypothermia. In this situation, you have to get your bag dry.
First, prevent yourself from getting any wetter. You have rain gear, that is good. Get into dry clothes, put your rain gear on, and stay active so you can keep warm. Sleeping in your rain gear is probably the best temporary solution if you're able to stay warm that way, if not, then you need to get your bag dry, which means bunkering down in a dry shelter. You only have a bivy, so you have to build something else if you can't find any natural shelters like a frost pocket or dry thicket.
After you have dry shelter, you need to start a drying/warming fire (see How to light a fire with wet firewood?) and hang your bag so it can catch the heat (don't hang it overtop of the fire). Build a reflector on the opposite side of your fire to reflect the heat back into your shelter and onto your bag. Now you have to stay there until your bag is dry enough to keep you warm.
Years ago I was caught in a totally unexpected and sudden blizzard/snowstorm. I started out the trip at 85–95°F (30–35°C) in August so I was totally unprepared for cold weather in the days when I owned a goose down sleeping bag only.
I hastily found a reasonably wind-sheltered site, pitched the tent, and dove inside as the storm raged. An hour or two later, I felt the trickle of very cold water flowing (head to foot) through the ground-contacting portion of the sleeping bag. Turning onto one side bought perhaps 10 minutes of relative comfort. After about three repositionings, comfort was unattainable. So I got out of the bag and tent—standing naked in the fierce blizzard—and wrung out the totally soaked bag of several gallons of water (or so it seemed) and shook it vigorously to fluff it up. It was not dry, but at least it was not soaked.
Getting back in the sleeping bag was heavenly: It was way warmer and I fell asleep immediately. However, soon it became soaked again and the cycle repeated throughout the night, something like once an hour. Not the greatest night's sleep.
By first light (04:30) the storm disappeared and I got out to look. There was perhaps eight inches of Cascade Concrete™ (snow) with drifts of more than two feet. My tent site turned out to be a small gully which was draining maybe 50 feet (15 m) of the PCT.
The temperature was chilly at about 10 °F (-12 °C) and my hands were too cold to do a decent packing job (why would anyone take gloves or mittens for summer backpacking?), so I basically stuffed tent and sleeping bag into—and onto—the pack. Though it was a three hour hike from the trailhead on the way in, it took more than seven hours to return because it was difficult following and finding the trail combined with the extra effort of walking in frequently deep snow.
After considering this misadventure for decades, my advice for a soaking wet sleeping bag is:
Your solution isn't a bad one. Adding whatever fleece you have under the raingear would help. Putting your feet in your pack will help keep them warm.
Much depends on other factors.
How cold is it getting?
I have routinely done trips with teenage boys in the eastern slopes of the Canadian Rockies in September. We have had weather as hot as 30 C (upper 80's) and temps as low as -12 C (~7 F) We used dacron filled bags nominally rated at -20 C (~0 F)
While it was unusual for a bag to get totally soaked, we found that if it did happen, wring it out, and crawl in. Wear your polypro. It's not a pleasant night, but you will warm up the bag and get some sleep. One person described it as "Making desultory love to a clam"
I consider September in mountains to be more dangerous than winter. The tendency is for ice water rain and/or sleet, making it harder to keep dry.
Keeping Dry
We used tarps -- the blue woven plastic ones from Canadian Tire. Each pair of students or pair of staff were issued with one 10x12 tarp and one 8x10 tarp, and a bunch of light nylon rope. First night we taught them kids 6 ways to pitch a tarp, depending on anticipated weather and wind. Then senior boys would check the rigging of the newboys to see they hadn't done something silly like leave part of their ground tarp sticking out where all the drainage from the top tarp would drain onto it.
Coupled with foam pads, kids generally stayed warm and dry. Most of the time there was nothing more serious than condensation where a bag was against the plastic.
Now that said: We were in areas where fires were permitted. We cooked on wood, and most evenings built a drying fire. (Do not give me static about fires please. I took my nephew into Willmore last summer in August, the height of the tourist season. We saw 8 people, all on the last day, 3 hours from the trailhead. Most trips I've not seen anyone else.)
This is country with LOTS of creek crossings -- at least one an hour, and on one notable stretch on Mumm pass, 57 crossings in 3 miles. Being well soaked from the knees down, and more than damp above, on a wet day was common.
This meant that a cold wet expedition could stack the wood high to keep warm.
Are you above timberline?
Timberline is associated with two subfactors: Often little shelter from wind. No wood for a warming fire. One of our policies on any trip that had newboys, was that we camped below tree line. We did alpine camping with the seniors -- mostly, for them it was their third or fourth trip. They knew how to keep from getting wet.
Reducing the consequences.
Ok, you're wet, now what:
Contrary to one of the other answers, I would consider moving at night under conditions that got you wet in the first place to be a poor idea.
Before you do this in an emergency, do it as practice.
These are all well put together answers. Here's an additional step regardless of the season
In an emergency a tea light candle carefully set on/in a disposable pie tin can heat a 4 season tent enough to aid the drying process - just be sure to isolate, ventalate, and insulate the flame properly! -- (tent model, size, wind, temp, and humidity will all change effectiveness)
Carying at least one mylar emergency blanket at all times is wise. In situations where you are wet these can save lives, and they bring some peace of mind while you are dry. I've always been comfortable after laying one loosely over my sleeping bag on uncomfortably cool summer nights.
Dollar stores often sell all three items, sometimes with emergency blankets big enough to turn into a sleeping bag.
Weight: A tea light candle ~3 grams. Disposable pie tin ~4 grams. Emergency mylar blanket ~12 grams.
Due to the weight to effectiveness ratio, I go through about 2 mylar blankets a season using them almost daily. They block the sun well too.
