This depends entirely on your VPN setup. Some VPN administrator require
that any clients connecting then route all possible traffic through the
VPN. It's not usually so much that the client's IP address changes as
much as it is the client machine is no longer allowed to be reached via
that IP address per routing rules. For the record, I hate these types of
systems because of this kind of pain/suffering.

There are other options I suppose, like once the first SSH connection dies
instead making a connecting via the IP address associated with that client
machine from the VPN setup itself (assuming that can be accessed).
Another option that I've used before is explicitly overriding the lousy
VPN client settings and adding the ability to talk normally to systems or
networks by adding routing rules. This can be easy (one command one time)
or harder (one command in a loop over and over) depending on how
persistent the VPN client is with detecting these kinds of unauthorized
changes. Using the 'ip route' command to add the route back to your local
network is usually a breeze, though.

--
Good luck.

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