Next OS after High Sierra 10.13.6 for iMAC 2011

You don’t. That’s as far as it can go. That model is obsolete.

4trees_Roma wrote:

I installed a 500 gb SSD - does that make a difference?

Only in how fast the machine boots, launches applications, and transfers files. It won't do a thing to let you run a more up-to-date version of macOS.

Since the version of Safari in High Sierra is rather old, you may want to install Firefox 115 ESR. It isn't getting new features, and Mozilla won't be providing any more security updates for it after September 2024.

Mozilla Support – Firefox users on macOS 10.12, 10.13 and 10.14 moving to Extended Support Release

It is true, a 2011 iMac is VERY obsolete and cannot run any Mac OS beyond 13.6! It is well past time for a new Mac.

A new HD or SSD will not change that fact.

4trees_Roma wrote:

I hear that iMac 2011 cannot go beyond High Sierra 10.13.6. Is this true also with an upgraded HD?

I installed a 500 gb SSD - does that make a difference?

No.

HDD or SSD does not make a difference. The 2011 iMac can still only run 10.13 High Sierra.