Disclaimer
The Author of this posting offers the information contained within this posting without consideration and with the reader's understanding that there's no implied or expressed suitability or fitness for any purpose. Information provided is for informational purposes only and should not be construed as rendering professional advice of any kind. Usage of this posting's information is solely at reader's own risk.
Liability Disclaimer
In no event shall Author be liable for any damages whatsoever (including, without limitation, damages for loss of use, data or profit) arising out of the use or inability to use the posting's information even if Author has been advised of the possibility of such damage.
Posting
There's likely some feature improvements in the later IOS version (e.g. maybe rapid-pvst exists in the later version but not the earlier). But even if you don't need feature improvements, there's often benefit to update to the last release in the same IOS train to get all the bug fixes.
Technically, believe you need a contract or a new IOS purchase to upgrade the IOS. As the box has long been end-of-everything, suspect you don't have a legal way to acquire the software.
For a definitive answer, you'll need to contact Cisco, but I also suspect unless you're a big customer of Cisco constantly buying new Cisco equipment, they might only recommend you purchase a new 3560X (with a possible trade in allowance).
