Check your local specialty hardware stores for teensy-tiny drill bits.  I got mine at Fasten-All, Ace may even have them.  Take a board with you, to check the size(s) you need.  Mine was kinda pricey ~$2.00, well, pricey for a tiny drill bit.  Or you may be able to find an assortment on Amazon or ebay.  Just twirl the bit between your thumb & fingertip to drill out the remaining solder after you use the braid or a solder sucking device.  I've had good luck. YMMV, of course.

73 Dan