Most metro areas will have anywhere from five to ten such organizations, each of which will have a different "style". The trick is to find out where they are hiding, as you have already determined.
The classic "big band" (five saxes, four trumpets, four trombones, four rhythm, plus any vocalists) may be a school affiliated operation (in which case look to be playing a lot of 40s through today "jazz-oriented" stuff), or it may be a privately operated bunch.
Both versions will work for money if they can get it, but the school ones are more based upon the experience of it all rather than money. Some of the school operations will let you play for free (although you won't get any credit hours if the group is part of a course at said school). And, some of the school operations get pretty far out there as far as experimentation with the "jazz idiom" is concerned. This appeals to some "pure musicians" more than it does to others.
One of the "problems" (if it is one) with a "big band" is that the repertory of most such groups will be focused on one particular era. That is to say that, depending on the person in charge, there may be a concentration on one particular type or style of music. There's nothing wrong with this per se, but it does tend to limit the appeal of the group in the final reckoning.
Looked at from a commercial standpoint, this is a big limitation, as the audience for the classic "big band" era is shrinking year by year. Many such groups play for far less money to entertain such crowds, and they are there more for the joy of playing than for the money. And, many such groups have "expanded" their repetoire into other eras in order to have a wider appeal.
If you are jazz oriented (as your tag line seems to indicate), then one of the school based groups would suit you well. However, keep in mind the instrumentation limitations here. You will far more likely find a seat in one if you play baritone or bass trombone rather than alto or tenor or trumpet. Some have huge rosters, but not all get to play every time.
Once you start talking about any commercial groups, you face a different process. If they perform for money, they will generally rehearse less, gig more, and have a limited call for new people. New people are going to have to measure up to the skill level of the rest of the group, so that may or may not work in your favor. (Doubling clarinet and flute is a big plus here.) And, when they do need someone new, it's usually going to be by referral ("Hey, I know a pretty good trombone player!" said by one of their players) rather than by any other process.
Another factor may be the repetoire that they use. I have had many excellent players show up only to decide that working for me was not what they wanted to do, solely based upon the tunes in the library. If you have a tenor guy who is really wound up in improvisation, he may not want to have to play vocal charts where the singer is the center of attention rather than the sideman. You'll learn as you play more what kind of music you would like, and your preferences may steer you in one direction or another.
No matter what the final solution is, the way to break the ice is to start by making contacts with others of similar interest. If you are a long term resident in the area (and it's hard for me to cypher out just where "UHHHHHH" is located), you have made friends with others in music over the years. These guys will be your best source for referrals.
If you are young, another starting point would be with your teachers at school (who often play on the side as well as teach).
If you get called to sub in a group, make sure that you are up to it technically, but also make sure that you are reliable. And, come prepapred to play whatever is asked of you. With classic "big bands' that may mean showing up with a clarinet or flute as well. Nothing bothers a group leader more than to rely on someone only to find out that they never show when they are expected, or that they can't do what they claimed they could do when it's go time.
Over the past few years, I've dealt with a lot of excellent musicians and vocalists. However, some of the most valued members of the groups with which I've been associated have been better people than they have been musicians. That's because (as people) they are more reliable than musicians who are far better than them in pure technical terms. Guess who gets the call when I need someone almost on a "right now" basis?
Put another way, the world's best tenor player isn't any good to me if s/he can't get to the jobs on time.
And, here's where I put in the pitch for the union. It is true that the union won't get you much work, but it does put you in the directory for your local, and it does put you in touch with other players who will put you in touch with other players and so on. It also will make you more hire-able for certain types of work if you are so inclined. That "hotel job" you referred to might be a union group, and if they are in a non-right to work state, then you would be obligated to join the union anyway.
In a way, it's a lot like making friends. It's not something that you can deal with on a cold and calculating basis. ("I'm going to get her as a girlfriend" seldom works, from what I have seen.) But, if you are around enough others who are sharing similar interests, the relationships will come in time.
When I relocated from the Saint Louis area (actually rural Southern Illinois), I went from a musical life that involved limited commercial work, participation in three local area orchestras and about four theatrical operations (pit orchestras) to an area where I knew no one. The approach that I took was to have one of the organizations with which I played (the Centralia Cultural Society) get me a list of similar community groups in the Greater (ha!) Houston area, then contact them when I got down here.
I started out playing bass clarinet for no less than six different community orchestras (rarer instruments always help here, either that or playing any of the strings), plus working (either gratis or for limited money for local theatrical groups. As time went on, I made a lot of acquaintances (I would hesitate to call them "friends"; for one thing, I usually only saw them through the agency of music), and they in turn would tell someone else: "I know a guy who plays baritone and bass clarinet and clarinet and the other saxes; perhaps you could use him?"
I also joined the two area locals of AFM, 65-699 (the Houston local) and 74 (the now defunct Galveston local). It's not required for most work down here, but it does put you in touch with a lot of other folks.
As in most of life, ability is important, but of more impact on the bottom line is who you know.