No you don't need to upgrade your app to a newer version of Jet (and no, there is no new version of Jet), you could still use that bits until they are supported by the Net Framework OleDb classes and the libraries present on the target machine.
The version of Office installed by your user has nothing to do with that.
Of course this library is outdated. The new bits for opening a Microsoft Access database in the new format (ACCDB) are called Microsoft.ACE.OleDb.12.0 and you can find the connection strings on this site
These new libraries allows to write application for 64bit but I can't really recommend to switch to a 64 app unless you have compelling reason to do that. For more info see this question