There's so much to choose from
Frankly, the one reason to go with 7.62x39 is the popularity around the world... for the same reason, going with the AK platform for that round makes sense (spare parts).
If you just want another caliber for your AR, decide what you want to do with it... stealth (300 black out or similar w/suppressor), get between .223 and .308? 6.8 spc. Carry fewer rounds with a pistol rifle combo... .22, 9mm, 5.7. Big power? .458, .460, .500, you can even get .50 BMG uppers (they don't quite fit the mag well

)
But 7.62x39 is about the one caliber for which, i just wouldn't bother.
From AR-15.com, most of these have been bolted onto the top of an AR-15 lower...
223/5.56 and based upon that case
6 x 45
7 x 40
357 Mag Rimless
300 Whisper / .300Fireball / 300-.221
6.5 x 40
.30 Coyote (30 x 45)
.17 Remington
.17 Mach IV
.20 Tactical
.19 Calhoun
5.7 FN
.204 Ruger
6.8 SPC
30 HRT
338 Spectre
10 mm Mag
35/40
219 WASP-C
7.62 x 40
5.45x39
7.62 x 39
.338 x 39
6 PPC
6.5 Grendel / 6.5 AR / 6.5 PPC Improved
22 PPC
7 sCAR ....
30 Carbine
5.45x39 / .21 Ghengis
7 BR
6.5 BR
300 Sabre
8mm (belt fed upper)
.308 (again a belt fed upper - no longer available)
458 SOCOM
44 Mag
440 CorBon
44 AMP
.475 Tremor
50AE
499LW
50 Beowulf
.50 BMG (single shot)
.50 Blackpowder (1 made)
9mm
45ACP
32ACP
22 LR
17 HV
10mm
7.62x25
224 BoZ
7.62x24
9x23
.357 Sig
.357 Auto
40S&W
223 WSSM
243 WSSM