Since you’ve already got one tree, sink a 4x4 post in concrete (or just bury it if you want less permanence and like to live dangerously) to use for securing the other end of a hammock (with one end wrapped around the tree).
The post can serve double duty too - you could use it as a tetherball post (better to use a metal fence post in that case though? Or do they make dedicated tetherball posts), for either you or dogs.