You are stuck in Liberty only for the first few missions - after mission 4 you have the run of the
whole Sirius sector. A small number of systems (about six) are too hot for any ship you can have at that point and you won't get the high-paying jobs that yield $20k or more, but otherwise you can do as you please.
I did exactly that in a campaign where I wanted to map every jumphole, secret wreck, base et cetera. I also took a mission out of every base that would give me one. When I went to story mission 5 I had visited over 40 systems and about a hundred bases, and flown almost a hundred bar missions. I also had 2 million credits in the bank and about 1500 kills - including Sabres, Titans and several dozen Falcons (because they drop powerful missiles that you are not yet allowed to buy at that time

).
BTW, you can get past the first four Juni missions in less than 3 hours, or 1.5 hours with T-Hawk's mod
Campaign Cutscenes Disabled. This would be absolutely wrong for the first time through the campaign since you would deprive yourself of a lot of fun, but for a repeat run (say, if you want to beat the campaign in a specific ship like the
Starflier or
Bloodhound) the techniques and tricks we have discovered for
Freelancer Done Quick can be very useful.