You are doing the right thing to plan a head. I found it easier these days to find an independent available than a few years ago (2005). It can be hit or miss. I recommend that you check cityvibe.com first, then check for the reviews for the last minute appointments on weekends. You also have out of town visiting providers that you can check out on eros, but make sure to check the reviews first. Don't forget to check the TER ad board(s).

I've many to be accommodating of weekend appointments.  I generally make appointments in advance, though, so maybe that's the difference between you and I.