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Tailscale has some integration with Mullvad. If you have a Mullvad subscription you can use their servers as exit nodes without dropping your Tailscale connection: https://tailscale.com/kb/1258/mullvad-exit-nodes

Outside of the particular combination of Mullvad and Tailscale I don't think there is any other way apart from switching between the two.

Maybe I don't understand, but the tailscale Linux clients definitely supports multiple accounts. I use that to reach multiple headscale networks and a tailscale one. No issues for me using it this way.

Not elegant or performant but:

You could have a exit node that is setup only for that vpn that advertises it's routes. So connecting to tailscale gives you access to that network.