Usually, you don't need a multimeter to solve these kinds of problems. But, there's a very real possibility you've shorted or opened a coil wire inside or lead wire coming out of the pickup, and while that can be obvious, sometimes it's not. So, instrumentation helps. You don't need a benchtop lab-grade Norton unit traceable to NIST; a $10 Radio Shack one will do.
But, perhaps just as handy might be a set of
test leads...
They're real handy, and real cheap. A set of 10 leads at the link above is $7. You can temporarily wire things up a number of different ways, test different caps or pots, jumper suspicious parts out, on and on. Gotta have 'em. Just be aware that they're not shielded or anything like that, so any temporary wiring you do with them is gonna be noisier than hell. Once you have a set you'll wonder how you lived without them.
Also, a tuning fork is a Good Thing. Tapping on pickups is a crude way at best of testing to see whether pickups are wired, or wired properly. Get a little tuning fork like one of these...
and you just bang it on something handy, then hold it over the pickup. You'll be able to tell exactly which coils are hot and whether or not your switches/volume/tone controls work. Plus, you'll have a frequency standard.
$6 at Amazon.