You have a 6' panel so I reckon you will need (at least) a 4' spur, this will give you 2' in the ground and 2' bolted to the wooden post. Treat the bottom of the new post and try not to have it in contact with the ground, then it will last a long time.
It is a terrible job breaking up and taking a lump of concrete out of the ground when a wooden post has rotted, that's why I favour concrete posts and gravel boards. If you like, you can use dark brown masonry paint to help them blend in to the stained woodwork.
When you have got the old concrete out of the ground, you may have quite a big hole. This will always be loose, so unless you have an easily compacted soil, you may need to go a bit deeper (but not wide) so that the foot of your concrete spur is in firm ground. If the hole is very big, you may need a longer spur or extra concrete. With a fence, it is especially important to have good support to each side of the fence. Lengthways is less important as they don't blow over that way. Wet the ground and tramp or beat it down hard afterwards, before the winter storms come.
for neatness, you can put a box form round the top of the new post so that there is not a lot of concrete visible through the surface.