How can I replace the short(screw connecting) wire for a middle-of-run outlet replacement?
Before, on each side, you had 1 wire going directly to a screw, and 2 wires joining a short pigtail under a wirenut, then the pigtail going to the other screw.  That's a silly way to  do it.  

Rearrange the parts you already have so that all the white wires and the pigtail are joined on one wire nut. Then run only the pigtail to the new receptacle.  So only 1 wire will be on each side of the receptacle.  That is a much better way to do that thing.  This will work best with red or tan wire nuts.  

What's happening here is that one of the cables (b/w/red?/bare) is coming from the supply / breaker panel, and the other two cables are branching in two different directions to go onward to other loads.  All 3 cables need to have their wires joined and the power must also go to a fourth place, the receptacle proper.  That's why 4 wires are involved. 

It's real common for less experienced folks to get tripped up in wiring methods like this.  Who really cares if you use 4 screws or only 2 on the receptacle?  Not I, and not electrons.  The important thing is all the blacks get connected together and all the whites.  (Mind you, that's not the rule everywhere, but it's the rule in this junction box today.) 

You cannot pull more wire out of the wall.  Do not waste what you got.