system
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human
[Star Wars] Were there other rebel-held planets or sectors? How did they avoid having the Empire show up and crush them? Why didn't they keep their main base in those places?

In the Star Wars movies, there's usually a "Rebel Base"...like on Yavin or on Hoth.  The Empire is quite clearly capable of showing up and crushing these bases when they find out where they are...they lose a Death Star when attacking Yavin, true, but the Rebels still abandon it later.  And the Rebels hide their main bases out in the middle of nowhere because they are trying to avoid Imperial attention rather than being confident they can defend the area.

But clearly the "Rebel Base", wherever it is at the moment, can't be the _entire_ rebellion.  They are getting ships and supplies from somewhere, which means they have access to manufacturing capacity.  And there's just got to be more going on if they expect to overthrow the whole Empire.  So what happens to those other planets?  I know Alderaan gets blown to bits for merely being sympathetic to the Rebels, but what keeps the Empire from knocking out the rest of the rebel supply line and organization?  And if there are whole "rebel sectors" that are too well defended for the Imperials to attack them, why aren't they putting their bases in those places?
gpt
They had bases on Hoth and Yavin because they were remote planets in remote places of the galaxy. But mostly the leadership stayed there.

For the rest of the rebellion, the galaxy is HUGE. The sparsely populated Outer Rim is where most of the rebellion hides. It is too big for even the Empire to patrol. Most rebel supporters also orginate from the Outer Rim.

As you said, planets basically can't openly back the Rebellion, as the Empire cracks down on that, HARD. Most of the rebellion isn't actively fighting, they are just sympatizers.  
For supplies, the Rebellion have got many wealthy backers, like the Organa family. Millitary hardware are plentyful in the galaxy. Particularily in the Corporate sector, they don't ask many questions to buyers. If you have the money, you can just buy it. Much of the Hardware is also stolen from the Empire.

The rebellion doesn't have the hardware to actively fight the Empire, as the Empire basically got the entire galaxy working for them. What they do is staging hit and run attacks on convoys, forcing the Empire to divert resources to escort ships.  
Much of the bigger hardware is also stolen from the empire during such raids. For example, the Nebulon-B frigate is an imperial-designed escort ship, but so many have been stolen that it is just as common to see in rebel hands.