I say "try on" a bunch of different chords. Everybody's fingers are different. Some people might surprise you with what they find "easy" vs. what they don't.
Just run through a bunch of standard chords with them. See what they "like".
There are a lot of easy barre or semi-barre chords that would broaden the scope. And you can always have they try to play a chord that's "close" to correct for a given song and see if they like it. (I still often play E7 rather than E just because I find E7 so easy to get to... and it usually works)
You've got them doing F, and that's not particularly easy (for a 2-finger chord). If you can do F... G7 should be a snap! A should be easy, too! Dmin might be easy, too if their fingers like to bend the right way. And get them playing with adding a pinky finger here or there. I learned to do that from watching somebody on YouTube.
You should be able to get them some representative of every chord letter to at least feel like they have a lot of area covered.
A, Amin, A7
Bmin (on C, E and A, don't play the G... or de-emphasize it), Bmin6
B7 (on G, C and E... could be hard... or not, I used to find it hard to FIND, but easy to hold)
^^^ if you can do that, Emin is just moved over a string
C, Cmaj7, C7, Cmin
D, D7, maybe Dmaj7
Emin7, Emin6, E7
F, F7 (or F7no5... don't fret the A string)
F#min is easy if you can play G7
Fmin could be easy depending on the player, and very useful
G, Gmaj7, G7
All of the barred 6 chords.
Before you know it, they'll be able to play 25 chords rather than just 3!
Give them this:
https://ukebuddy.com/chord-namer and just tell them to play around with their fingers on the fretboard. If they find something that's playable and sounds okay to them... figure out what it is! Is it a chord? Is it a standard chord? Or is it some funky "jazz chord"? Either way, you'll find a use for it someday.
One thing that helped me was poking around on a uke song list that let me sort by key. I don't THINK in keys at all, but a key will use a particular set of chords. And once you find a key that often uses a set of chords that you're comfortable playing... you can look up songs in that key (or transpose to that key, I suppose) and keep jamming on the same chords! The Am, F and C would hint toward the key of A Minor, which is one of my faves.
This is a good resource for that kind of search. Also lets you sort by number of chords. (stay away from David Bowie and Supertramp, among others)
List of 1300+ songs for ukulele with chords. Each song fits on one page for easy playing.
scorpexuke.com