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Presently each set of 20 results we get costs us and we want to make the results better and better.

Oh I see, ok - I think you could improve it by providing different search filters or sort orders. At least that way we can stand a better chance that the 20 will contain what we need.
Also, in case you are unaware, the main search bar in steempeak doesn't work at all for me in firefox. I entire a search term and hit enter and nothing happens. If I click the magnifying glass after entering a search term, I am taken to the search screen but my search term has been lost.

Yeah the search bar may have a couple issues with usability most people think that magnifying glass initiates a search... sadly it does not.

So the search bar doesn't currently have a use other than to do a lookup for users?

I guess that's probably true. The top search bard is only a lookup of accounts. Wow that seems kinda sad when you put it that way. We should make the UI better or even the wording so that people know the top search only finds accounts.

Seems there's a difference between this simple search and our new advanced search found on it's own page.

I would probably remove the search input field totally for now and just leave the search icon so that people don't get confused or let down... then maybe update the search page so that it takes search inputs and so can be served as a result of the user running a search via the search bar in the traditional way.
if you make the search page take a search input you can also add steempeak to the list of searches supported by duckduckgo for example - which is pretty nice.