I have dug out stumps before and that's one job i hate to do.....When i first moved on this property i dug out about 20 stumps, and last year i made my garden bigger & had to cut 11 trees & dig out those stumps.
The stumps that i dug out was popular & wild cherry....The way i do it is start digging all around the stump, and when i see a large root i take an axe & cut that out, so I'm not putting so much strain on the BH jacks....and it helps so the tractor don't jerk all over the place
After i have dug down and removed dirt & the big roots...then i take my BH bucket and try to get under it from the back side, and see if it moves much....If it doesn't then i have to dig deeper and cut more roots....Then try to pull again & just work it back & forth and it will finally come out.
It all depends on what type of tree it is....(AND) all stumps are different when digging out....I have seen some roots on populars stay close to the top, and then the next one have roots that go straight down....and the wild cherries are a different story, and hard to dig out.
I have dug out stumps before that had such a big root ball on that i had to pull them, because they would not fit in the bucket.
Like Vic said, just take your time & don't rush it.