Wonderland is basically the central hub/in-between place for traveling between worlds, which is why there is a gate and other precautions. The defenses did their job perfectly up until the Curse hit: keeping anyone from accidentally stumbling upon the portals, and keep anything meaning harm from entering or leaving other worlds.
While Jumpers (I’mma gonna go with capitalized for now…) would be capable of some other magic if they knew the right focus words, their specialty—with practice, of course—is usually sniffing out portals. If experienced enough with portal magic, they could also choose which one they came out of on the other world’s side, but there’s always only one in Wonderland. However, in Miles’ case, if he’s a drunk, it would have diluted his senses, then add in that all Jumpers are fourteen years out of practice, and, well. Also, being a Jumper wasn't something one had to be born into, but it was common to be, so whether or not he grew up traveling the worlds, or started some time later in life is entirely up to you. Either way, and no matter what world he originated from (they pop up through out all worlds), he would likely be at least fairly familiar with Wonderland.
Before the Curse hit and all portals sealed themselves to protect the other worlds from infection (and, as a consequence, cut any Jumpers in other worlds off from each other), Jumpers had two main rules they had to abide by, rules upheld by both the portal magic and a sort of council of Jumpers with the job of watching out for anyone who may manage to get something past the portal magic:

1. It was impossible to bring anything from one world to another that that world did not already have in some form.

2. Telling any denizens of another world who were oblivious to the existence of other worlds was forbidden. However, this did not stop them from spreading stories about the other worlds or being overheard talking about them to each other, which is how Earth—which was often a favorite of Jumpers for how much it tended to change—got many of its fairy tales and legends, as well as even some terms.

Because Wonderland was the central station for traveling, Jumpers used it as their own sort of base, where they could freely share stories of other worlds with each other, showing off battle wounds, etc. Since it’s technically an “in-between” world, neither of the rules fully applied there. Most people knew about the portals, but your average person couldn’t access them even if they wanted to, and Jumpers had more leeway as to what they could bring to Wonderland from other worlds, allowing them to trade goods with each other that could come in handy in other worlds. However, they were still watched carefully, and had a system for keeping track of who brought what to Wonderland, and making sure the items were only sold to other Jumpers. Any violating that was severely punished by Wonderland’s Jumper council and the White Queen.
The last thing I can think of right now, is that they would always use at least one alias, depending on how “famous” they wanted a name to be among their fellow Jumpers, along with another reason that will ultimately get mentioned well before our guys travel.
Okay, I lied. This is the last thing. There’s one aspect all Jumpers share: an obsession with world-jumping. While anyone with the right powers can get into the field with the portals, and travel the worlds, once a true Jumper gets a taste of portal magic, there’s no going back. Though there are some exceptions, they usually can’t stand staying in one world for too long, and feel a “call” to visit another one. Quite a few of them went more than a bit crazy when they realized they were cut off from doing what they felt they were destined to do. They tried to reopen the portals, but all attempts failed, and the “council” on Earth basically disbanded due to a mix of casualties from toying with portal magic, semi-insanity, and the rest backing out and doing their best to adapt.
Let me know if you have any questions, or suggestions!