Heh yeah, an overdrive might make more difference. Teles - brilliant as they are - aren't 'heavy' guitars. They can do heavy tones pretty well, particularly if you stick a humbucker in the bridge (like a dimarzio Hot Rails), but a huge proportion of the tone you get comes from the other end of your rig: the amp and pedals.
Get a Seymour Duncan Lil 59 or JB Jr single-coil size humbucker. Great, versatile pickups and you don't need to get a new pickguard or anything.
- Fender, Taylor, Martin, Ibanez, Ramirez, Marshall, Boss, Morley, Mesa/Boogie, Univox, Shure, Monster, Dunlop, Seymour Duncan, DiMarzio, Lace, Sperzel, DW, Tama, Zildjian, and a little Johnnie Walker