The classic 3-saddle Tele bridge is notorious for poor intonation.
Your Telecaster deserves better.
Replace those saddles.
A set of compensated telecaster saddles, in any material your like (brass, steel, aluminium, etc), will greatly improve the intonation. The adjustment style differs between manufacturers, but generally each string will have its own compensation on top of the overall adjustment from each saddle’s intonation screw.
A set will be supplied with intonation screws and springs, so it is as simple as removing the old set and fitting the new.
Gotoh In-Tune Saddles are a favourite and are available as a set or on their replacement bridges. Wilkinson have a similar set available that’s a little bit cheaper and it’s worth looking at the Joe Barden offering also.
Although it may not be as accurate as a 6-saddle bridge, this upgrade is much cheaper and quicker to do than replacing the entire bridge unit.
Read more:
Upgrade your Telecaster with any one of these 5 guitar parts – Top 5: Telecaster Upgrades