Grew up in Fire ant country, Ocala Florida. They just love the soft sand down there. Anyways, never seen them climb a tree. Not saying they never had, just I've never seen it. I'd say you'd probably have more problems with the Argentine and other sugar ants that will climb anything and anywhere to get some goodies.
Maybe some kind of barrier like a lid from a coffee can with some sticky stuff on it. I'll bet as long as you keep food out of your hammock, there shouldn't be any problem. I've never had any ants crawl over the suspension to GET me. I have had spiders and other weird bugs fall out of the tree into the hammock with me...that's why I like a bug net!