Have you considered a .357 Magnum? When hunting, it has more more knockdown power and you can still shoot .38 Special cartridges in it for practice, smaller game, etc. You might also be able to find a used police revolver in .357M at a really good price.

If you Google search hunting deer with a handgun, you can find lots of articles and reference information that will give you more information. Here's a link to a Chuck Hawks article that addresses this very issue.

http://www.chuckhawks.com/handgun_hunting.htm