First, there is Dissidia 012, where Chaos, Cosmos, and Cid of the Lufaine are drawn through the Void from World A to World B, from which point Chaos and Cosmos pulled various warriors there through the Void. The Void is also how Shinryu got there in order to draw strength from the warriors of many worlds. I know that there are those who wouldn't consider Dissidia canon, although I would argue that is since one of it's main objectives is clearly to flesh out the story of Final Fantasy 1; fortunately, it's easy enough to point out a character who has appeared in more than one main series game (as opposed to the Cids who merely share a name) - in fact, I already have.

I'll start with Gilgamesh's description from Dissidia, as it's the most explicit about his dimension hopping. It lists his appearances as FF5, FF8, FF9, and FF12, before going on to say, "A vagabond who roams the Void and collects exquisite weapons. Formerly Exdeath's right-hand man, he has been travelling to
different dimensions since his encounter with Bartz." When he shows up in FF8 he mentions something about having to find the 'Dimensional Interval', and in 12 with a bunch of familiar looking, but cheap knockoff, weapons. In his appearance in the 20th Anniversary release of FF1, the first thing he says when you encounter him is, "At long last! The ultimate sword... is... MINE! Now I can finally go home!" And finally, although I'd take it with a grain of salt since he breaks the fourth wall at the start of the fight, in his DLC battle in FF13-2, he shows off the new weapons that he says he picked up in the rift. On the subject of that battle however, it is worth noting that the Ultimania guide for that game lists the locations he acquired the weapons he uses after he tosses aside the guns - Excalibur comes from a hidden room in Alexandria, Muramasa was recovered from the ruins of Zanarkand, Masamune was fished out of Doma's poisoned river, and the Zantetsuken is apparently not Odin's sword in this incarnation, but rather a nail shed by Chaos - and as far as I'm aware, the Ultimania guides are always considered canon.

It's also interesting to note, that since Gogo banishes himself to the void in FF5, and then the Gogo in FF6 looks almost exactly the same and has the same job, it's possible that Square had this idea for connecting everything through the void early on.

Finally, although it doesn't have to do with the void, it's worth noting that Square has flat out confirmed one other direct link between two games in the main series. Kazushige Nojima, the scenario writer for several games in the series (most importantly for this point, FF10, FF10-2, and co-writer for FF7), has stated that Rin would fund Shinra's efforts to refine energy from the Farplane using the Vegnagun's remains, but failed. His descendants, a thousand years later when space travel was developed, would travel to the Planet from FF7 and found the Shinra Electric Power Company - sourced from the FF10-2 Ultimania guide.