You're right, Tionese quality gear is the lowest quality of endgame PVE gear, but you probably only need to worry about upgrading it if you want to do group endgame PVE stuff (HM flashpoints and operations). That said, I think it would be rather difficult to improve your gear much without doing group content. It's been a while since I've done the endgame PVE stuff, but I think the only new (and better) gear that you'll be able to acquire without any group play at all would be the Black Hole stuff by earning black hole comms through the new space missions. Even the weekly mission in the Black Hole area on Corellia (that drops BH comms as a reward) requires you to complete a Heroic 4 mission (although this is solo-able for certain classes).
You can always buy higher quality augments, other item modifications (barrels, crystals, armorings, etc.), and some pieces of gear (implants and earpieces) off the GTN. But I think this would be a much slower process than acquiring the gear through comms and drops you can earn actually doing the group content.