Sorry I've got so many questions. I had a big browse of guitar shops yesterday, and I've been thinking a lot!
Probably the first thing I'll do to my tele is put some better pickups in. I'll play it for a few months as is first, and then decide what I'd like to improve about the guitar. But I'm guessing the pickups will probably have the most direct result on the sound.
I was looking at Seymour Duncan pickups, and the guy in the shop pointed out this pickup: it's a single coil/humbucker hybrid, and you switch between them via a new volume knob with a push/pull button.
Cutting Edge - Little '59 for Tele ST59-1b - Seymour Duncan/Basslines
I know it's not relic at all ... the Seymour Duncan site lists it as 'Cutting Edge', so I don't know if this would be going against the flow...