We never mind paying corkage or buying a nicely selected bottle with appropriate mark-up. That said, it's fun to get a pass to bring a nice bottle from home.
Foundry on Melrose for the first two bottles. Providence may still have no corkage on Mondays.
A good, but not always updated list can be found here, but always call before you go:
If you're searching for free corkage each and every night, consider a drive to downtown Santa Ana -- a long hike for you, to be sure -- to The Playground. The restaurantt's being discussed on a subsequent thread in this forum.
The Playground thread:
Lots of smaller places allow BYOB, even though it's not legal. If you are looking for inexpensive then try some of the Thai places in Hollywood's Thai town, like Sanamluang Cafe, Sapp Coffee Shop,
Pa Ord Noodle and the like.
If you want more upscale then places like The Foundry or Osteria La Buca have corkage free certain nights (usually Monday's to get people in their seats on traditionally slow nights). Check first about number of bottle restrictions if you are planning to take multiple bottles for dinner.