Month to month rentals are few and far between other than the option Steve suggests. Pets make it even tougher. My advice is to look at regular rentals and try to negotiate a 6 month lease. Would probably be no more expensive in the long run.
Bernard Gibbons, J. Rockcliff Realtors
DRE License # 01331583
Phone (925) 997-1585 - email@example.com
With former owners being displaced due to distressed situations rental rates are on the rise and Landlords can afford to be picky as far as the lease/rental duration; which, potentially may make a mo-to-mo rental relatively undesirable for you (due to relative cost) and the Landlord (due to the uncertainty and "extra work").
If as you search you find this to be true consider negotiating a mo-to-mo agreement with an "extended stay" hotel like the one in San Ramon below (which allows pets; $25/day until you reach $150; negotiate this as well).
Best of luck with your search and move.
If you would consider downtown waterfront area of "Benicia", I have a gorgeous water view townhouse/condominium that will be available apprx. 3rd weekend of July.
It is apprx. 1800+ sq.ft., 3 bedroom 2 bath, 2 car garage about 6 years old......owner may consider
a small dog or ??
Rent will be apprx. $2900.
Haven't started adverising it yet, so let me know as soon as possible if you have interest.