I can't say if this will work without knowing enough about your stump and fence. How far under the stump have you dug? If you keep digging deeper and deeper and get that taproot loose, then the stump will be able to rock back in forth in all directions. Then maybe you can move the stump to an angle where you can chop at the other roots.

I find using a mattock for digging is nice, as you can really get through the hard dirt that has adhered to the smaller roots you will pass on your way down. Sometimes it's necessary to have a very wide hole to get deep. For a 1 foot diameter stump I have often times used a 3-4 foot diameter hole to get all the way down there.