ideas to remove broken fence post?

A bit confused here

Dig down to extract post or dig a new hole beside old post and install a new post?
Ideally the post would be down at frost level possibly cemented in?

Now if it's an anchor post (corner) the best way is to dig down to frost level put 4 cross beams on the bottom about 3-6' long We than throw a tire over the pole to put more surface area at the bottom and bury it all.
It's how we hang our gates and they have not sagged in 40 years. So must be done right

We did get some faulty fiberglass hydro poles and have been using those lately but normally we use cement posts.

But if it's your white picket fence in town negate all the ideas I mentioned other than digging with shovel