If I remember correctly, "Need for Speed Carbon" has free roam.
On PSP it's difficult, I don't think there are games with open worlds like Zelda on Switch in which you can traverse large areas running or on horse; maybe you could try "Phantasy Star Portable 2", it has nice and different areas to explore in every mission (free roam in the sense you can take your time instead of rushing to mission objective).
Same for "God Eater Burst", it's like MH in mechanic, so not really open world but different areas you reach from a hub. Post-apocalypse graphics, ruined cities and so on.
"Dissidia 012", in the campaign it has something that resembles old world maps of the first FFs, you should look for gameplay video though.