The only weapons that allows you to hunt for both big and small game as far as I know in CA are-
Bow with various type arrows
Shotgun with slugs and shot
Centerfire Handgun (357 or 44) with bullets and shot loads
(may be limits on shot size that restrict handgun shot loads
on some species such as turkey).
The Natives ate fish and pine nuts. Then
they went down to the valleys in the winter. The fish are mostly gone, and the pine nuts vary from season to season. The Jefferies can make lots some years.
There are squirrels. The mule deer are mostly gone because of
habitat loss and too many people. A few grouse and quail
and rabbits.
For squirrels, rabbits and grouse either a 22 or a shotgun.
Quail require a shotgun.
For mule deer a center fire rifle or handgun, or a shotgun with
slugs.
You could expand your deer season with archery, as they
let you hunt during gun season with it too. And you can take any game with a bow (legally).
I would suspect the only game you will get is squirrels unless
you have done some scouting of big game beforehand.
In other states you may be allowed to use centerfire on squirrels or grouse. I like subsonic loads in my deer rifle
for this--use Trailboss powder, see the Hodgdon site for info.
http://www.hodgdon.com/index.html