Does yours still have the pickups that came in it from Fender? If so, they are Texas Specials, wound hotter than what some folks want for country.
You might try playing some other Teles you can find in guitar stores. Try the pickups (bridge especially) in any of these you can find:
1) AV '62 Custom;
2) Classic Series '50s or Esquire;
3) Classic Series '60s, '69 Thinline or '72 Custom (bridge pickup);
4) AV '52;
5) Lite Ash or Koa Tele.
The AV '62 Custom Tele is in my experience hard to find so good luck, but the pickups in it are supposedly a good bet for clean country. They are A5, the bridge is staggerpole with a traditional baseplate.
The Classic Series Teles are easier to find. A5, the bridge is either flatpole or staggerpole depending on the model, but doesn't have the traditional baseplate.
The AV '62 Custom bridges may vary from one ot the next, I've heard reports of them being in the mid 6s and others of them in the mid 7s. In any case they are rated brighter than the Texas Specials. The Classic Series are mid 6s, very bright, less harmonics/mids than the AV '62 due to no baseplate.
The AV '52 is A3 flatpole with the traditional baseplate. The wind and magnets probably puts it tonally somewhere between the A5s I listed above and the Texas Tele pickups. If you find the AV '62 and the Classics too bright, this might suit you better.
The Lite Ash and Koa Teles have Duncan A2 pickups. One Tele I had this bridge in seemed real smooth, smoother and less bright than the AV '52 A3; but in another it seemed brighter. In both though, this pickup was more tonally complex than A5s or A3s, responding more dynamically to different height settings, volume and tone control settings, and pick attack.
You can find any of these pickups used on ebay, you can find the AV '52 pickups new as well.