Why not just use a bigger pin?
I had the same issue, so what I did was get one of those large paperclips(with a diameter that matches the hole in the shaft, not the head), and cut several new "pins".
To install, I held the new pin with needle nose pliers, and heated it with a lighter for a few seconds, then pushed it in through the aligned holes in the head and shaft. The heated pin slightly melts the plastic around the hole for a tight fit(makes the hole in the plastic the same size as the pin). No more slop.
It does eventually wear the pin out, but leaves the hole the same size. Paper clips are cheap, just make a bunch all at the same time for spares.