There used to be an option in FireFox to open search bar results in a new tab but somewhere along the line that was removed. The installation on my Desktop had somehow held onto that preference through the upgrades I'd made until recently when another plugin I installed changed that.
Thankfully the essential Tab Mix Plus extension came to the rescue.
If you've got it installed, go to Tools -> Tab Mix Options and choose Events from the top. Now select the "Search bar" check box from the "Open tabs from:" group. You may also need to change the options under the Links page of those options to ensure that they're opening in a new tab (or window if that's you're preference).