I purchased a home recently and I am just getting around to fixing the 6 foot privacy fence. A few of the posts appear to be loose in the ground and the fence is leaning fairly substantially around these posts. I dug down to the concrete footings and there does not appear to be any rot. I would assume that the reason for the fence being loose is the posts were not dug to sufficient depth or not enough concrete was used for the initial pour. Are my only options to break away the existing concrete, dig a wider hole, and repour the concrete?
The posts may be loose for a variety of reasons and it's worth spending a few minutes considering which are likely so that you can mitigate them. As you said it could be that the posts were set too shallow or the concrete (and the hole it was poured into) was too narrow. Both of those point to the same thing: weak soil, meaning soil that can't bear much pressure. It could also be that a previous fence post stood in the same place; somebody removed that post and added dirt to make the hole smaller but did not compact the dirt well. Water problems can cause subsidence or soften the soil. Wind can put a lot of force on the fence, causing the post to work like a lever to compact the soil around its base.
Yes repair is going to require replacement of the existing post and concrete. You don't have to break the concrete in-ground though: it can be hoisted out intact with the help of a tractor, a chain hoist or come-along hung from a tripod or scaffold, or if it is loose enough a simple lever may do the job.
Breaking the concrete while it's still in the ground often results in extra soil being removed to provide access. That makes a wide conical hole.. and a person is tempted to replace some of the disturbed soil rather than fill that yawning hole with concrete.. and soon there's a loose fence post again.
If you determine that the fence needs more support than the existing post and concrete had provided then dig the hole wider and/or deeper to provide that.