Reddit reviews Traveller: Spinward Marches (The Third Imperium) (Traveller Sci-Fi Roleplaying)
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Used Book in Good Condition
Mongoose 1st edition had a line of "Third Imperium" supplements that seem exactly like what you're looking for.
This is probably the best single book for what it seems like you're looking for:
The Spinward Marches
It nicely summarizes up the 3I setting in the context of one corner of it - the Spinward Marches sector. There are other general 3I supplements that are not sector-based: Starports, Spinward Encounters ( okay, a bit sector-based, but easily portable ), and Sector Fleet.
There are other setting books in this series: Deneb, Reft, Alpha Cruces, and Trojan Reach sectors, the Sword Worlds and the Solomani Rim. There are Major Race books for Aslan, Zhodani, Droyne, Vargr, and Darriens.
There are also a few adventure books in this series, set in the Marchers.