I am trying to construct a compact hyperbolic surface tessellated with hyperbolic right-angled polygons with $n \ge 5 $ edges. I found quite easily a way to do it for $n$ even, but the odd case seems more tricky. For example, if n=5, my idea was to take two right-angled pentagons, with edges say $a$, $b$, $c$, $d$, $e$, and glue the couples of edges $(a,a)$, $(c,c)$ and $(d,e)$ preserving the orientation, and $(e,d)$ reversing it. My guess is that I should obtain a torus without a disk, that is, with one boundary component. However, I cannot see the tessellation on the torus, which makes me wonder if my construction actually works.
If you glue four regular right-angled pentagons (side-lengths $\ell$) at one corner, you get a right-angled hyperbolic octogon with alternating side-lengths ($\ell$ and $2\ell$). You can glue either along pairs of opposite long edges, or along pairs of opposite short edges to get the figure you want.