What methods could possibly be used to achieve time travel?

In threads like these, I always introduce Kip Thorne’s ‘temporally-displaced wormholes’ method. Kip Thorne is the bloke who advised on the movies Contact and Interstellar.

That is the only method I’ve ever seen that would work reliably, and can guarantee that you don’t get lost in time and space. (If only you can manufacture wormholes in the first place).

1/ build a wormhole, with two connected mouths maybe ten metres apart. You can walk through the wormholes and pass from one mouth to the other without passing through the space in between.
2/ send one wormhole mouth off on a relativistic spaceship, travelling a circular path several light months or lightyears long. Time dilation will cause the two wormholes to be temporally displaced from one another. The amount of displacement depends on how fast and how far the spacecraft travelled.
3/ place the two wormholes back on Earth and walk through them in either direction. If you walk in one direction you go to the future; if you go the other way you go to the past. If you want to travel into the far future you can just keep waling through the 'hole in the same direction over and over again till you get to when you want to be.

This method of time travel would get round Pardel-Lux 's problem of rematerialising inside the Earth or in outer space; the two mouths of the wormhole would be both safely on Earth, and you couldn’t go anywhere else. You could, however, go elsewhen.