New fence, existing poles cast directly into the ground.

[topmount]

Considered this as well, but will the new concrete adhere well enough to the old if you don't remove all the wood?

Well, it doesn't really need to adhere either, it's not going anywhere. Just lay a layer on top too.

 

[stugfar]

If it had been me: drilled into the wood and carved/chiseled away the remnants with just the right amount of precision and filled in with new concrete. An orderly person might have connected new and old concrete with rebar. I might have done that in a couple of exposed spots. But it's a fence, not an oil platform that needs securing

 

[stugfar]

Doesn't it get wobbly with those pole shoes for a fence?

Well, not necessarily, but true that flat iron might be preferable.

 

[crazytok]

Post base on a stick becomes incredibly wobbly!

Here are a couple more alternatives...

1. Dig and cast new footings
2. Maybe you can dig out a bit on the sides of the footing. Saw off part of the concrete with an angle grinder but leave the post in place up to ground level. Then attach a flat bar on each side with thru bolts or lag screws, which you then use to fasten your new post in place?
3. Slutagräv.se, screw down new footings so you don't even have to dig.