I always have a pistol where allowable in the woods (not in state parks)...if for no other reason than to dispatch snakes. To answer your question it is allowable ALL the time without a CHL.
NOPE, YOU CAN'T WITHOUT A CHL DURING BOW SEASON if hunting whitetail or mule deer and turkey!
Archery and Crossbows
(see Artificial light in Restricted Methods section)
Only the archery and crossbow equipment prescribed in this section may be used for taking game animals or game birds. Archery and crossbow equipment may not be used to hunt deer during the Muzzleloader-Only Deer Season. It is unlawful to be in possession of a firearm while hunting with a broadhead HUNTING point during the Archery-Only season, except a person licensed to carry a concealed handgun in Texas may carry a concealed handgun.
NOTE: A firearm may be possessed in camp, in a motor vehicle, or while hunting lawful game other than whitetail or mule deer and turkey (e.g., exotics, feral hogs, squirrels). http://www.tpwd.state.tx.us/regulations/fish_hunt/hunt/means/