The quickest method is to ask other people. Either here or on a site like WhatTheFont's forums or Typophile.com's Font ID forum.
If you want to just get better at it yourself, I'd suggest hanging around the above mentioned sites and gain exposure and experience to the number of typefaces out there.
However, in terms of just being a more efficient graphic designer, I'd suggest that the problem is that you have 4000 fonts.
That's kind of like a carpenter having 4000 saws. That's just a burden. You have to get an extra trailer just for them. Have to sort through them at the job site. It can be a real pain.
A carpenter likely maybe needs a dozen saws, at best. Crosscut, rip, a few Japanese options. Hack saw. Coping saw. Then a handful of power saws and a few backups.
Plus, there's the issue of costs. 4000 quality saws likely cost as much as 4000 quality typefaces. Either you have a budget of 400k to purchase quality tools, or you have to sacrifice quality and buy everything from Northern Tool (the typeface equivalent of downloading everything from Dafont)
Typefaces and fonts are tools just like that. You want a nice set of quality tools, but not buckets of old rusty cheap stuff from Northern Tool. :)
So, I'd suggest you begin paring down your font collection. My suggestion is to find:
- 6-10 workhorse text faces for body text (serif and sans)
- 6-10 workhorse display faces for headlines, logos, etc.
- 10+ alternative options (more unique faces you need often).
Then maybe 50-100 'other' faces you grab as needed.
Instead of 4000, you want 40. Too many options isn't good for design. :)
I'll leave you with the words of the late, great Massimo Vignelli:
The only thing that’s important to understand is: when to use it [a typeface] and when not to use it. Or what to use.
And then, good designers can come to it. They come very well along with very few typefaces.
And very good designers doesn’t use more than few typefaces.
And when they are less good the number increase, and if they are worse they use all of them. laugh
Now, Vignelli took this to an extreme, of course (It's been said he only ever used 12 typefaces) but he has a good point.