If memory serves me correct, it's pretty simple...
Somewhere in your "Index.Templet" or "Index" files for your forums is where those buttons are located up top. You know, "[Forum],[Rom Map],[Ram Map],[Search], etc."
All you have to do is find the html and/or css script for those buttons, duplicate where you want the button, and place something like phpBB3/search.php?search_id=newposts after the <a href=" part.
In theory it should work, that's what I did with my old forums, Syndicate Underground, but that was a Simple Machines Forum which is a little different than phpBB3, but not by much.
Chances are those buttons are calling on a css script and you'll have to find out where it's calling to, and then modify a "New Posts" button in. It shouldn't be very difficult, it just sounds like it.
The only thing that worries me is that I put "phpBB3/search.php?search_id=newposts" after your URL and it didn't work, usually it should. But perhaps it just needs to be in the html page itself, IDK.
'tis been awhile since I've messed with forums.