Although you mention being in the suburbs, if you are an hour walk from a road, this is effectively a wilderness first aid situation. This guide to wilderness first aid of a snakebite seems to cover the topic pretty well. It doesn't seem that copperheads require much special advice, except that your odds of a "dry" or non-venomous bite are higher than with many other venomous snakes, so the outcome is more likely to be favorable. Go to the hospital even if you think your bite was dry.
The advice from the Snakebite Foundation guide seems excellent, so I'm going to 100% quote it below (because sometimes links go dead) rather than try to summarize it. One thing I will add, is to make sure you know the most effective way to describe your location, such as the names of the trails in that park, and well-known landmarks. Plan for this in advance, and have a map of the park with you (either hard copy or saved on your phone). Some parks have their own park ranger service with an emergency number that they recommend calling them instead of 911 because they will be better able to understand where you are and how to find you. Find out if that's the case, and make sure you have that number in your phone.
WHAT SHOULD YOU ACTUALLY DO IN THE EVENT OF A SNAKEBITE IN THE
BACKCOUNTRY? The following steps are what you actually should do in
the event of a snakebite in the backcountry from any species of
venomous snake. Ultimately you are going to need antivenom because
antivenom is the only definitive treatment for a snake envenomation,
but the tips below are designed to help you get to the hospital alive
and leave the hospital in one piece.
Carefully walk backwards and find a safe space to sit down nearby
before the venom drops your blood pressure and you pass out and hit
your head. Many snake venoms disrupt blood clotting and the last thing
you want is to cause internal bleeding on top of your snakebite.
Remove any rings, watches, bracelets, and anything else that could
become a tourniquet if your limb swells up like a balloon. These items
can be very difficult to remove once swelling has occurred, so
exercise some foresight and remove them right away.
Circle the site of the bite with sharpie and write the time next to
it. Mark the edge of the swelling and pain, make a list of your
symptoms, and repeat every 30 minutes or so. Always record the time
next to each mark. The vast majority of snakebites can be diagnosed
and treated by your symptoms and severity of the envenomation without
requiring a positive identification of the species responsible. That’s
why this is so important!
...then mark the leading edge of the swelling and pain (separately, if
needed) and repeat the process as they advance up the limb. It is
particularly important to record the time that swelling reaches and/or
passes the various joints on the affected limb.
If you begin to experience signs of anaphylaxis (swelling of face,
mouth, or throat; hives; difficulty breathing, etc) use an epinephrine
autoinjector (EpiPen or generic) if you have one and then take
Benadryl and Zantac. If you don't carry these things in the
backcountry you should do yourself a favor and get them because you
can't MacGyver an EpiPen out of nothing. A lot of things can cause
anaphylaxis and EpiPen can also be used as a last-ditch intervention
for severe asthma attacks.
If you have cell phone reception call 911, tell them where you
are, when you were bitten, and the list of current symptoms you just
wrote down. If you aren’t in the United States, look up the local emergency services number (whatever the equivalent of 911 is) and add it to your phone before you head out.
If you don't have reception, plot the safest and most expedient
path to find it or reach a vehicle (whatever is safer/faster) and then
start hiking out.
Time is tissue and it may be better to walk yourself out in an hour
than to sit on your butt for 5 hours until a helicopter can show up. I
think the idea that one should do everything possible to avoid
speeding up circulation of venom is bad advice. You are already
terrified from being bitten by a snake so your heart rate and blood
pressure are already sky high. I've treated lots of bites in remote
places and pretty much all of them had to hike out to reach the
hospital. Figure out the fastest, safest route to find help and then
make it happen.
Source: HOW TO SURVIVE A SNAKEBITE IN THE WILDERNESS, snakebitefoundation.org
- Don't use a tourniquet.
- Don't try to cut or suck out the venom.
- Don't apply ice or a cold pack.
- Don't try to catch the snake.
- Don't take any NSAID analgesics (aspirin, ibuprofen, Aleve, Motrin, etc.) for pain control.
Note: The "copperhead" snake in question here is the North American species, eastern copperhead (Agkistrodon contortrix), not the Australian copperhead. First aid for an eastern copperhead bite should follow standard snakebite first aid practices, which means do not use a pressure-immobilization bandage for an (American) eastern copperhead bite. Don't follow advice for bites of the Australian copperhead, which is a neurotoxic elapid. Australian copperhead bites may be helped by a pressure-immobilization bandage (but only if you fully understand the correct way to do it; incorrect use of a pressure-immobilization bandage is worse than not using one at all). The linked page includes instructions for pressure-immobilization bandages, but I omitted them here to avoid confusion.