Since .NET doesn't support loading more than one version of the same assembly into any Application Domain, it's not going to be running in the same application pool. With some trickery you might be able to get things working using two separate virtual directories, one hosting the old solution, one hosting the new.
As long as your application doesn't rely on Session state that might be the simplest thing to get working. As long as you then make sure both applications have the same MachineKeys configured. It'd be a sort of "Web farm Deployment".
If you are using Session State, you might need to configure session to be stored out-of-process in the state service or in SQL Server.