On my boat, which is older than yours (1967), the tiller head is connected to the rudder shaft by a key way or key insert that had badly worn down and allowed a lot of slop in the steering. Replacing the key way/insert entirely fixed our problem.
Since you say that this area is not the problem let's look further.
Is this your boat model? Hunter made several flavors of 25 footers: HUNTER 25 sailboat specifications and details on sailboatdata.com
This spec says it is a skeg hung rudder. There should be a lower bearing or shoe for the shaft to rest in. Perhaps there was a lower bearing there that has been worn down? I had to make a bronze bearing for our boat for this exact usage.
The worst case scenario is that the shaft has become 'loose' from the rudder itself which would suck. You said you tested for this but could not detect any independent movement of the rudder from the tiller/rudder shaft. Have you noticed if your rudder takes on water? If it is taking on water it will drip or weep for some time after it is hauled for the winter. Rusty weeping water is definitely a bad sign, if you notice it.
There should be metal tabs or straps welded to the rudder post that are encapsulated by the rudder laminate. If water gets in there the metal straps can corrode and if they fail the rudder can simply rotate on the rudder shaft - which is obviously not happening with your boat.
In any case, you should try to tighten up the rudder/tiller connections if it is bothering you. I do like the idea of using strips of plastic milk jug material for packing or bearing for the rudder shaft and tube. Another option is to make rudder shaft bearings with West System epoxy using their graphite filler (I think JimsCAL linked this).
Good luck however you proceed.