You will find these useful. I always have.

Add to Search Bar {web link} Make any pages' search functionality available in the Search Bar

Context Search {web link} Expands the context menu's 'Search for' item into a list of installed search engines, allowing you to choose the engine you want to use for each search.

I'm not aware of a way to easily reset the default search engine after you have changed this. When you close and restart Firefox then you could use the user.js file to set a specific search engine. You can change the default search engine in the search bar with command + cursor Up/Down. This search engine will be used for all search bars (location bar, about:home, about:newtab included).

I'm a little confused about the difference between default and the search engine you used last. The whole idea of adding "one-off" buttons to the panel that drops down from the Search Bar was to allow you to easily send a search to a different site without changing your default. That isn't available on the address bar, so I don't understand how your default is getting changed or how you are choosing a different search engine.

Could you describe your workflow and, if relevant, any extensions you are using to modify the address bar behavior? Maybe that will help us come up with suggestions.

Also, you may be aware of the option of assigning a short keyword to the search engines you use most, for example, g for Google, t for Twitter, etc. Then starting a search with g mozilla will direct the search to Google. If you prefer a keyboard solution, this might be the fastest way to direct searches to non-default sites.

To get the "old" dialog where you can assign keywords, please copy/paste the following into the address bar and press Enter/Return to load it:

chrome://browser/content/search/engineManager.xul

By "default" I mean default: the one it defaults to unless I specify something. The way Firefox works now, it's like there is no default, it uses whichever I last specified in the Search box.

I'm calling "active" the search engine whose icon shows in the little search box. If I type in the address box, Firefox sends my search to the active search engine.

I use the little search bar when I need to look for something on a specific site. Otherwise, when I do a general search over the entire web, I want to just type my search terms in the address bar and have it use my go-to search engine, regardless of what is active in the little search bar. This is what I do probably 80% of the time.

Yes, I can still use the little search bar for this, as long as I reset the active search engine first.

I investigated y'all's suggestions and discovered the solution: An extension called "Keyword Search" https://addons.mozilla.org/en-US/firefox/addon/keyword-search/ What it says it does is exactly what I want it to do, and in my testing (i.e., the one time I tried it) it worked as promised.

Thanks for the ideas

Hello,

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errorprone said

By "default" I mean default: the one it defaults to unless I specify something. The way Firefox works now, it's like there is no default, it uses whichever I last specified in the Search box.

Perhaps are reassigning your default using Cmd+down arrow. To avoid changing the default, just use the down arrow without cmd to select the search engine for a one-time search.

Or maybe you reverted the search bar to the "old" design from the days before Firefox 34. In that bar, yes, selecting a different search engine always changed your default search engine.

Anyway, the Keyword Search extension sounds like a good workaround for your needs.