I would check with the airlines directly, probably United, Continental, and Southwest. A handling company may give you an airport job and some experience, but no airline benefits.
I think you can check CO
/UA under United Airlines and it will specify which carrier the job is with. They are still separate companies in some ways, until the merger is complete.
I think a week or two ago I saw UA
hiring, but didn't see anything when I checked yesterday. I would check the sites at least weekly.
Find out which carriers do their own handling and apply for those first.