Hi Wolf,
I think you might need to clear the ARP table on the switch and PC.
You can check if the ping request for the non-pingable ip is even reaching the ASA (current def g/w) by running captures on that interface, or running the "debug icmp trace 1" command ("un all" to stop).
If the ping request is not reaching the correct ASA, i would suggest checking ARP values.
Also, if you could elaborate a bit more on the topology, with ip subnets and all, it would give more insight into where the problem may lie.
-Shrikant