Without a huge tree digger, you should dig a foot away from the stump and soon you find a good size root, follow it, removing the soil and loosening the root, if you get some 3 feet of roots you are fine, make sure to make a sharp root pruning. it helps it restart the rooting. You should find at least 3 or 4 branches of roots... then the others you just prune them.
At my tree, it is impossible to try to remove dirt with it, my soil is like gravel, you dig and it breaks... if yours is the same, have some buckets of water, soon you dig enough, water pour water and it helps you find which way the roots go. I just dug up my black madeira that i had grafted onto my tree's roots. it was interesting. I did not find lateral root for at least a foot from the graft, but found lots of arms and more ramifications. with water, I was able to get several thinner roots.
Best luck.