Ultegra 6700 10-speed shift cable frayed at the lever end. I got the lead head end out far enough to cut it off, then pulled the cable out from the other end. There were several 1/2" bits of cable left at the head end, which I removed. When I try to install the new Shimano cable, there is a lot of resistance, causing me to think there may be more bits of cable stuck in the routing path. Any suggestions on how to proceed? I tried inserting the cable from the other end, but it doesn't follow the routing path and comes out the front of the shifter. Thanks.

You sure you got the right cable thickness?

Yes, same thickness. I have tried using a piece of the old cable too (at a clean cut) and it won't go in either.

Beside buying a new casing, I don't know then, sorry.

If the cable comes out in 'the wrong place" (ie 'the front of the shifter') when inserted backwards, if I understand
what you did, then you may have inserted it into the brake cable pathway. I agree with your diagnosis of a strand
of cable left in the shifter. You could cycle the shifter through its full range of motion for shifting repeatedly and
see if some bit of cable trash comes into view and try the reverse approach with a freshly cut cable again through
the shifter channel, if you can identify it. Try it in both directions.

Sorry, there is only ONE way to insert a cable in a shifter. The fact that you tried inserting it "the other way" indicates to me that you don't know how the cable is supposed to be in the shifter.

My guess is that you put the cable in wrong.