Just look for a FT job. I used to be of the mindset that it was always good to have your hands in multiple cookie jars so to speak but over time my perspective has changed somewhat, especially in these turbulent times. While yes, having privileges at multiple facilities does increase your leverage and allow you to experience multiple environments. The reality is that most jobs have a relatively low opinion of PRN docs. You are there to fill gaps in the schedule but are not valued like a FTE. The FT docs will always preferentially get their requested shifts and there is always a vested interest by the employer to hire FTE until you aren't forced to use PRN anymore. Nobody wants to rely on PRN on a consistent basis because they are by nature...inconsistent. They might could give 6 shifts one month, 2 the next, none the next, and so on. They don't feel particularly loyal to a job site because they are working the majority of their shifts somewhere else. They can be entitled and demanding as well as difficult to schedule at times.
Another way to look at it... Whatever shifts you are working PRN takes away from your full time gig. If you are working 16/mo and only 12/mo at your FT site, then the employer will staff accordingly and sooner or later when your PRN jobs dry up, you won't be able to pick up extra shifts at your main site because they staffed so that there wouldn't be any extra shifts/mo left over. That's why it's better to negotiate extra hours in your contract that are guaranteed instead of minimum hours. My colleagues at my current gig are contracted 120 hours. I negotiated for 140 and probably would have negotiated higher if they had let me.
Anyway, working multiple PRN is a good thought. The multiple environment thing is not untrue but in today's environment I really don't think it's the way to go. You should search for a great FT gig and take it before someone else does. If you truly are lost about what types of environments you want to work in then take a firefighter job with one of the major CMGs for the first year or two to figure it out.
I used to do PRN a lot outside my FT gig for the first 3-4 years outside of residency and stopped for all the reasons above. Plus, it's a pain in the ass to stay credentialed at multiple other hospitals. The only thing that would make me change my mind would be if there was the ability to scalp high paying gigs at nearby EDs on a consistent basis like several years ago but those opportunities are gone for the most part.