Mirror lenses are built something like a telescope, and comparatively inexpensive. We can't get a small bird to occupy a major portion of the frame with any less magnification. There are many good choices in that range. A word about f-stop and your lens. F-stop refers to the amount of light coming through the lens.
Low numbers mean more light, high numbers mean less.
F16 allows less light through than f5.6. The maximum amount of light a lens allows is measured in terms of f-stop and is a crucial factor shooting in heavy shade or shadow, and at dawn or dusk. Get what's affordable and be philosophical. A 2X teleconverter, for example, doubles the power of a lens. As a reference, the optical magnification at 500mm is about the same as 8X binoculars.