Hello and welcome to the forums,
Currently to manage this feature you'll need Administrator access to your epesi.
When browsing records, search keywords entered are matched against fields currently displayed in the table.
If you want to search by other than default fields, you need to show them in "Table view". To do so, go to Menu->Administrator->RecordSets, in field "Record Set" select "Contacts" at the top of the screen and switch to "Manage fields" tab. Next to each field you'll find "Edit" button, that allows you some basic property modifications.
Please make sure that by enabling more fields you won't loose clarity of the table, consider hiding (disabling Table view) for some other fields instead.
I should also mention that due to complexity of some fields, searching by these fields is disabled (one of such fields being Zone field, I can elaborate on why this field is not included if needed). You can see which fields you can search by simply by clicking "Advanced Search" while in browse mode. Each of the fields that you can search by will get an input input box, those you can't search by won't get one.
If you need to limit your results by Zone field, I'd recommend enabling Filter for that field. You can do this the exact same way you change "Table view" for a field, just use the "Filter" checkbox while editing fields properties.
After that in browse mode, in filters (top-right) you can filter your results by zone.
In the future, we have plans to move table-view management to user-side, allowing to change columns displayed, columns order, width and filters, with preset saving with easy access and way to switch between them.
Hope this helps.
Kind regards,
Arek