You should be able to migrate from S5.3.x to S6.2, I don't see it saying you can't in the install guide when migrating from an existing appliance. You must be running S5.3.x, not S5.3. You are correct, it's not possible to go to S6.0 because that is for the third-generation TCS appliance only as you've noted.
It's not easily possible to bring the TCS back online after you've run the migration assistant tool. I've done it myself a year ago when I migrated our hardware TCS servers to virtual, but it wasn't easy and I don't recall how I did. If you need to bring it back online now, I suggest you contact TAC.
The migration assistant tool should have moved all of your TCS S5.3.x data to a shared drive so that you can migrate it to the new virtual server when you bring it online - media, database, logs (pretty much everything that is on the E: drive will be copied). After you get the VM TCS up and running, you should be able to run the migration tool to copy the TCS data to the new S6.2 server.
Can you tell us what you have done to this point, do you have a S6.2 TCS up and running, per the TCS 6.2 VM Install Guide.
Note: Your question would be better off in the TelePresence section of the forums, where these devices are more actively discussed and people can help get you the answers you need. You can move your question to the correct section by editing it and changing the categories at the bottom.