I've spent a lot of time discussing and pondering Classic Traveller. While this post applies to all brands of Traveller it deals mainly with the sandbox approach: you make up your own setting (as in the 1977 edition LBBS) and set aside the well crafted Imperium. It can also be applied to creating space opera settings of any stripe.
Using the Imperium or not is the Big question for many Traveller referees, the single most important question about their TU. A great many gamers of CT choose to set the Imperium aside and create their own governments/societies from scratch. Some people profess hate for the 3I though most I've encountered just prefer their own homebrew. in discussing all this I have come to likethe original sandbox approach but using the Imperium or a variation on it is far from the only tweak you can impose. In fact in my opinion the Imperium is not a huge issue and does not invalidate the sandbox setting.
Traveller, according to the original editions, is supposed to be about adventurers on the frontier. It owes more to The Man Who Would be King than Star Trek. While exploration can take place (and should) it's usually exploration in concert with some other overriding goals the player or party has (credits, power, shiny tech.) Since you're on the frontier where the law doesn't go very far naughty PCs like you can move more freely. So what doe the Central Government matter? If anything most stories revolve around people trying to get by without the Government lending a helping hand or grabbing them by their necks and throwing them in jail. It doesn't matter if an Emperor, President or Rockstar is issuing edicts.
I'll explain it in terms of tomatoes. The question that matters is not who tells you how much you can sell your cargo of tomatoes for, it's how far you can get away from price controls by the Ministry/Department/Executive of Tomato Management.
Of course that does not take into account the local government. Then again you got a ship and a jump drive: find a local government that lets you do what you want (or can't stop you.)
Setting aside the Central Government (often known as the Imperium) question; what does matter to players trying to earn a credit (or steal one)?
One of the biggest changes you could make from the LBBs would be to legitimize psionics, make it a part of the society. In this case you'd have to figure out how ordinary folks get by without psi. Are they defenseless? Are psionic shields available? Did the psionics become the 1%? If psionics are relatively common does every adventuring group need a psi to do anything? Or are the adventurers on the frontier where there are no psionic institutes to make their fortune?
Aliens are another major tweak. There's nothing on aliens in the LBBs and I think that went a lot of the way towards the human primacy theme in the Imperium setting that followed. Humans were all over the darned place, ran the biggest government and their worst enemy was other humans (psionic no less!)
What if the aliens are the bog deal have all the high tech and lord it over humans who are barely capable of starflight? It gives a very different feel to the setting.
For your consideration: piracy. As it stands piracy is an encounter now and then and a lost cargo (if you and the pirates play it cool). What about a frontier composed of hostile pocket empires and clinet states where privateers and pirates are a way of life? If you want a lawless setting full of adventure (read appeals to people with poor risk assessment) read up on the settlement and warfare in the Caribbean.
As a final exercise I propose the following setting.
The Aliens (Hivers work well for this) have ruled the stars for millennia and are not going away soon. They used economics technology and social engineering behind the scenes to keep any human polities small and weak. Fear mongering and espionage assures the region's pocket empires hate and fear each other and all employ raiders and pirates as proxies. The one glimmer of hope is the psionic institutes formed under deep cover to analyze and develop human psi potential and equalize the tech imbalance. Psis work to recruit others of their kind and begin a diplomatic mission to unit the many branches of humanity. Throw in pantropism and androids if you want.