Players are often seen challenging out calls/non-calls by the linesmen, but when the chair umpire makes a call correction, can a player still challenge the corrected call?
Absolutely, I've seen that exact scenario happen many times. It usually results in the other player having to request the challenge since the call gets changed/overruled. Challenges are available for any call, whether it's a linesperson making the call or a chair umpire.
See VII. The Competition PDF here. Look at page 130 - Section VII, #7 where it says
The original call or overrule will always stand if the electronic review is unable, for whatever reason, to make a decision on that line call or overrule. In this case, the player shall not have his available number of challenges reduced.
specifically mentioning the linesperson call or the chair umpires overrule.