I am replacing a rotted out light pole in my yard. The underground feeder that runs to the light pole is only long enough to run halfway up the new light pole. Is it acceptable to splice an extra length of cable to run to the top of the light pole? I don't know if this is acceptable since there will be a splice hidden inside the light pole.
No -- splices need to be accessible in case something gets messed up and the splice needs re-doing. There should be a space in the base of the lamppost to make the needed splice/connection, though -- look for an access panel as part of the base.
Code says that splices have to be accessible, although I can think of two exceptions: My well has a splice between the direct burial line running across my yard to the 3 strands of wire that go down the bore. This splice is just in the riser pipe that brings power from the burial trench to the well cap. There is another splice at the pump. Those are covered with plastic waterproof compression seals.
Normally I would expect the base to have a built in junction box for splicing. After all, you may have multiple lights on that circuit. If it's a home made light this may not be the case.
Another possibility is that it's low voltage lighting. Code is a lot more liberal for that, since the risk of electrocution is much lower.
If you choose to ignore code, and splice anyway, I would suggest using crimp connectors and heat shrink tubing.
Yes, you can just splice it, but if possible don't just twist it and tape it - that's probably asking for a short sometime in the future, it would be best to splice and solder it and tape it.