Upgrade path - Oracle Standard to Oracle Enterprise
It's easy because internal database structure almost identical:


Backup all database files and /network/admin (just for sure); 
Uninstall Standard Edition;
Install Enterprise Edition;
Point ORACLE_SID to old database;
Startup database and run catalog.sql and catproc.sql from new version;
Recompile all invalid objects.


Note that software version, release number and patch level must be identical.