There isn't only one method, there are a few. Partially the choice depends on which canoe you have (big and wide, very stable or tippy, or a small solo) and if its calm water or not. What you can do with a big canoe might need some tweaks when it comes to some small solo and a canoe with good secondary stability will be easier to get into. Extra flotation is not always needed, its useful to have for practice but some materials allow the canoe to stay at the surface (my cedar canoe will float even if completely swamped for example). When you flip a canoe air remains trapped underneath so, in that condition, it floats upside-down and you can flip it back up without swamping it completely, also canoes that would sink when swamped often have built-in air chambers at the ends so in some ways its possible to selfrescue without extra flotation.
(In the other question you linked you are thinking a lot about extra flotation, it can be an help, mostly because whatever room the flotation bag take is less water to bail out, but is not absolutely necessary in a canoe unless you are in whitewater to not get swamped, and there once you flip you don't want to stay close to the canoe, so no re-entry. The other point is that you cant rely on flotation, you might not always have it in the canoe. I don't use extra flotation unless I will encounter rapids, in normal canoe camping trips I used it only once and one of the bags developed a leak so was useless. If you learn to re-enter on your own without any other aid then having extra flotation is just more safety)
Its not a great idea to re-enter a semiswamped canoe and start bailing, the idea is that the gunnels are lower on the water so the re-entry should be easier but I find it leads to an almost completely swamped canoe which can capsize on you again and, if you make it in and there are waves, you keep getting in more water than you can bail out.
The first thing to do is to flip the canoe upright: from underneath, lift one side first to break the vacuum and then lift and throw it at your side, its the Capistrano flip (easier than what you would think even with heavy canoes and it doesn't leave a lot of water in the hull), or you could roll it upright and start bailing while floating beside the canoe.
With two people you would re-enter towards the middle, one person on each side of the canoe, one stabilizes the hull while the other climbs in, first from the water then from inside, or both climb in at the same time thus balancing the hull.
When alone some hull shapes will let you climb in from the middle but usually its best to re-enter a bit towards the ends so that your weight on the gunnels will be closer to the centerline.
The movement is one smooth motion, grabbing the gunnels with the hands, you want to put your chest inside the boat and then the feet, then its all a matter of rolling on your back and do whatever you need to do.
These two youtube videos show you how its done:
Someone teaches to float horizontally at the surface and hook the gunwale with one foot and one arm, the intent is to "lift and roll" into the canoe, this never worked for me though.
Use a stirrup you hang at the gunwales to help pulling yourself in, I used it once during practice and it works but it always gave me a bad feeling (like if my foot could get trapped in the stirrup if I didn't make it in). It is however useful if there was an injury (like a shoulder injury received while bracing improperly before capsizing), to use the stirrup its more like kicking back rather than stepping up, the mental image is to pull the canoe under you with your foot and not to climb onboard like it was a ladder on a dock (or you would end with the feet under the canoe capsizing it again).
Use a paddlefloat like the ones used for kayaks, the paddlefloat can be an extra item in the boat or made using a spare, or one's own, lifejacket. I find it laborious and its an extra item to look after, besides it wouldn't work if the paddles are gone when you flipped over. And using your own lifejacket works when everything goes according to plan but if you fall back in trying to re-enter you wont have a lifejacket on
use the cargo, if you have packs, in the water to stabilize the hull either as float on one side or as counterbalance on the other depending on the situation or, if the pack is heavy and you wouldn't be able to pull it back in afterwards then use it as counterbalance inside the canoe
All in all re-entering the canoe is not that difficult. Its like picking up an heavy cedar-canvas and rolling it up on your shoulders for portaging: it just requires practice.
My cedar canoe is 18ft, narrow, tippy and rather heavy (north of 70#) and we did practice re-entries solo, two people and even two people plus packs and a dog. What seemed really difficult at first has gotten easy with practice. The Capistrano was easier than what I thought from the beginning even when solo and packed, its more technique than strength (and you need to break the suction first otherwise you wont lift even a light canoe).
Videos and explanations aside its one of those things that benefit from having someone teaching you, it fixes mistakes right away and you wont take days to learn.
Here there are another couple videos from youtube:
PS: flotation bags inside the canoe wont help you to not flip it, if you don't know how to get in in first place you risk flipping it again, remember that you don't have any secondary stability to count on in a swamped canoe. If the weather is bad and there are a lot of good waves it can become a very long and tiring bailing fight. Its not a good situation to be in. Also, somewhere on the net there was (or still is, not sure) a well known webpage promoting inflatable sponsoons outside the canoe, those differ greatly from the narrow and long "sausage" sponsoons you see in some touring canoes or the builtin sponsoons of the old cedar canoes, they work against you when you have to flip the canoe upright and in waves they can lead to capsizing again.
PSPS: if the selfrescue is going well, you are in the water with waves and wind but the canoe is not swamped anymore and you learned how to get in, sometimes an useful thing is to go upwind at one end of the canoe and lower that end in the water using your weight. The other end will lift and the hull will slowly windvane. You can use that to avoid having waves hitting you broadside (If you decide to swim the canoe to shore instead put the hull in between you and the wind/waves)