I have a freeze proof faucet on my house that leaks back through the shaft
at the handle. I replaced the rubber seal at the other end. That isn't the
problem. It's the packing but I was told the packing in these faucets can't
be replaced because they are pre-formed. Any thoughts on this? Can I leave
the old packing in place and wrap additional new around the shaft and
tighten it down?

have you tried just snugging the nut down on the packing? if that doesnt
work adding some packing may do it. isnt the packing just a rubber
washer? if so,you should be able to get a replaement lucas