Take all the money you've saved by rescuing an old bike and get a whompin' bright light for the fog. I think I'm going to upgrade to Dinotte's 400L front & rear setup.

If the fog is thick enough, you might run into problems with a bright front light and glare blocking your vision. Remember that cars need to run on low beams instead of high for that reason. Most bike lights are like car high beams since they just shine everywhere without a horizontal cutoff.

If you ride often enough, seriously check into hub dynamos. There have been some really nice lights appearing recently, including proper optics that don't send half their light into the trees. At the moment, I'd get a B&M Lumotec IQ Fly for myself. Keep a lookout for a new one from B&M called the Cyo, too.