I went through GT: Rim of Fire, and while there's not a huge amount that's new, there were some additional details on known polities and a few new things. All things in quotes are directly from the book. Things in parentheses other than locations are my speculation/extrapolation from what's in the book.
IISS first entered Solomani Rim around 100, contacting the four major powers (Old Earth Union, Easter Concord, Dingir League, and Vegan Polity) soon after. By 200, trade was occurring between the powers and they were beginning to explore nearby Wilds. Easter Concord in particular is called out as working with IISS to open up new worlds. Easter Concord also most willing to join TI due to having remnants of RoM noble structure (other polities did not). Most early contact is via scouts, since Imperial diplomatic and military interest is directed spinward against Reaver and Aslan states. EC joined TI in 426, DL in 582, VP in 586, and OEU in 588.
Terran Mercantile Community – trade alliance rather than actual government, mostly in Sol and Arcturus. Extended from Lagash to Mirabilis, which would give it some territory in Dingir and Capella as well. Began contracting around -1500 under pressure from Reaver states to spinward. By -1400 trade collapsed to a level that ended piracy due to a lack of targets. Became Old Earth Union in -1110.
Old Earth Union – extended from Ys to Barnard and Dismal to Sirius. Hostile to Dingir League and Vegan Polity. Attempted to conquer Near Bootes Cluster in -400 and failed. Incorporated into TI in 588.
Vegan Polity – survived Long Night “almost unscathed,” but like most Rim nations was isolationist. (It's unclear whether the ten worlds within five parsecs of Dingir were part of the League or the Polity. I lean toward the Polity.)
Easter Concord – settled Quaver (Solomani Rim 1110) around -500, 11 parsecs from Easter. This is slightly outside the EC’s borders from other sources.
Arcturus Federation – created by Banasdan, stretched from Altiplano to Anenerkuk and had “sporadic contact” with Vegan Polity. Fought a civil war around 200, joined the TI in 583. (Probably nine systems total – Altiplano, Noricum, Banasdan, Aqilat, Nyarlathotep, Arcturus, Jael, Tunguska, and Anenerkuk – the wiki leaves out Tunguska, but it’s the only J1 route to Anenerkuk from the rest of the Federation.)
Bootean League – formed around -400 in response to Old Earth Union aggression. Comparing the League to the modern Federation, it did not include Aquitaine, but did include Vantage, Polyphemus, and Hamilcar. (Speculation: it likely covered the eastern part of the northern J1 network of the modern Federation, southeast to Aristotle and northeast to Chernozem. Saxe is also likely, but it possibly didn’t include Sequoyah, since that planet did lose Jump and its history before the -200 civil war is unclear. Excluding everything from Sequoyah west on the J-1 main, Teucer (a recent colony), and the southwestern cluster around Aquitaine that’s J-2 from the rest of the Federation, the League may have been 10 systems. This size or smaller would be consistent with it not being considered a power on par with Easter, Old Earth, Dingir, or Vega.)
(Esperance may not have maintained Jump on its own but should have had at least some access to Jump-capable ships, since settlement by Vegans “continued throughout the Long Night.”)
(Iilike may have had Jump, since it maintained a “buffer state” between the Dingir League and Old Earth Union. This would likely need to include Markhashi, since that system is more directly between the League and the Union than Iilike is, due to Sirius being part of the Union and the League extending down to Enki Kalamma. Shuruppak and Markhashi would be the shortest route between those, with Shuruppak-Iilike the second-shortest.) Iilike joined the TI in 596.
(Lagash may have maintained Jump, since it’s noted as having had conflicts with the Old Earth Union during its expansionist periods. The closest Union world was Barnard, 7 parsecs away via J-2 routes, and those conflicts probably took place around Agidda and/or Nusku.)
Scandia retained spaceflight and contact with Terra throughout the Long Night, but maintained its independence. (This implies deep space refueling posts, since there’s a 3 parsec gap between Scandia and Dismal.) It didn’t join the TI until after the Rim War, having gone from independence to Solomani in 740.
Hathor was able to settle Anenerkuk during the Long Night,. (Presumably it kept Jump, since settling via sublight ship, while possible, would be unusual enough to warrant a mention.)