It’s really about where your “input focus” currently is, not really about if the Find window is open or closed (but of course closing it shifts input focus to a better place in this case…). If focus is in the Find window, all the usual shortcut keycombos that you are used to while in the editing window are NOT active, so pressing them (e.g. F3) does nothing.

But all is not lost! You really don’t have to reach for that mouse! When input focus is with the Find window, the Enter key does the Find Next that you seek. As a bonus, in recent versions of Notepad++, Shift+Enter will do a Find Next in the opposite direction.

But perhaps you will not like that it is a different keystroke than your brain desires – Enter when you think F3. It is (sort of) an issue because it interrupts your workflow to have to remember context in addition to the function you want to execute…a similar necessary brain context-switch was discussed here–you may be interested in reading that…