You can do it in most cases and it should be understood: reparler, remonter, revoir, reprendre, recourir, relire…
but it will sound weird in some cases, even if it makes sense: raimer (aimer), se révanouir, rehaïr, repréférer, recomprendre…
and in some cases it means something else entirely: réviser, reporter, remiser, répondre, ressentir, regarder… and in this case, adding another re- will sound awkward.
So I would advise to do it in informal context, and especially orally: native speakers do it all the time, I have even heard « J'ai re-ressenti » and « J'ai re-regardé » ; but you shouldn't try it in formal context, or in write it if you are not sure of how it sounds. Note that it should be ré- before a silent h as in réhabituer and in some cases before a vowel réarmer, réapprendre.
For more insight on using re-, le wiktionnaire has good usages notes.
Now if you are not sure or don't want to use re-, in most cases, and especially for dynamic verbs, using encore is a good replacement in composite tenses
J'ai rejoué → J'ai encore joué
while de nouveau sounds right in simple tenses
Je rejoue → Je joue de nouveau