How can I add extra field in "wpsl-search" area ?

If you open the wpsl.gmap.js and search for this line.

$( "#wpsl-search-input" ).addClass( "wpsl-error" ).focus();

You can add something like this below that line ( untested ).

$( "#wpsl-search-wrap" ).after( "<div class="wpsl-custom-error">My error msg</div>" );

Then in the else part of that code section you remove it again with if the search input is not empty.

$( ".wpsl-custom-error" ).remove();

You can style and position it with the wpsl-custom-error class.

You do need to minify the code from the wpsl-gmap.js after you made any changes on a site like this https://javascript-minifier.com/, and place it in the wpsl-gmap.min.js before it will show up in the front-end.