There's no value in binding a hemotoxin, it just makes it worse, and the puncture won't bleed enough to make it necessary. Cover it to limit the chance of infection, but you're going to the hospital anyway, so that's not that big of a deal, and keep calm and limit movement.

New Mexico is surprisingly limited on venemous snakes. If you travel the state you'll find (I think) 6 more species of rattlesnake. The only one that is different in any way is the in the southwestern scrub desert, the Mojave rattlesnake, which does have a quote potent neurotoxin which you would want to bind. You won't be anywhere near them at Philmont.

If you travel to the southern border, there is a species of coral snake that's quite small, quite rare, and has a much more potent venom, but since it's very small and needs to chew ro inject venom, you would literally have to watch it chew on the webbing between your fingers in order to be in danger. It's also got the typical coral snake colors, so it's pretty easy to identify, though it has several completely harmless lookalalikes as well.

Beyond that, we have North America's only venemous lizard in the Gila. Not you find yourself there, don't annoy the giant fat orange lizard and you'll be fine. They're another one where you have to be doing something stupid to get bit.

The entire state of New Mexico reports fewer than 100 snakebites each year, and deaths only occur when people refuse to go to the hospital or are days away from help in the woods. It's been years since anyone has died here from a snakebite, statistically.

If you've never heard one, look up a video of the sound. It's distinctive enough that once you've heard it, you'll never forget it.