if solder joints are hard to access you can slip heat shrink over the wire. some wire was rubber and certain radios have a lot of it. the cloth covered stuff stands up better. If you like using the heat shrink it comes in colors and you can smoke it a bit with a lighter to make it look not so bright. In the old days they didn’t use heat shrink but rather something called spaghetti. It doesn’t shrink, just slips over the wire and you can get new old stock stuff if you look hard.
that one isn’t too crowded some of the rogers sets and others have a lot of old rubber wire and it can be a time consuming affair.
to not get confused, take a couple of bits of colored insulation and stick them over any wires you clip. that way if you are drawn away by the phone or somethign and go back you remember where they went. taking pics helps too.
as stated depending on color it can turn to mush or peanut brittle.
its a bit overkill but you can get meter lead wire. it is very soft ruber but also good quality wire, better than you need but sometimes I like it because it seems ot fit in well and easily available.
insulation has ratings so the wire insulation should really match or be of higher grade than the voltage it sees. most old radios didn’t use plastic wire. at least until the 50’s or so.. I don’t use that some do, be careful of possible heat sources that might melt plastic if you use it. soldering iron will melt it of course.
you can also get a covering , found in stoves. it is heat resistant and not shrinkable, usually white. its ok to add that if protect if you are near a point that might be a bit warm and you just want a little further protection. If you can, try to keep the colors as they are sometimes referred to by color.
some like to re-stuff the old paper shells and keep all the wiring looking original others feel it cant’ be seen anyway, Your choice how far you take this, might depend on the value or age of the set or how fussy you want to be.