Yes, you can safely do multiple jumps before refueling.
If you use the Galaxy Map to find a multi-jump route to a system, the solid lines indicate how far you can go without refueling and the dashed lines are past the point where you'll need to refuel.
When you're in a station with outfitting available, you can go into Outfitting and look at the specs on a couple internal modules. Your Frame Shift Drive will have a maximum amount of fuel it can use per jump, and your fuel tank will have a maximum capacity. That will tell you how many jumps you can make before needing to refuel.
For example, the Sidewinder and Hauler both come with a C2 fuel tank that can hold up to 4T fuel, and an E2 Frame Shift Drive (FSD) that maxes out at 0.6T of fuel per jump. (4/0.6)=6.67, so with those ships, you should be able to do 6 jumps at max jump range before needing to refuel. More jumps than that at less than max range. Both of those ships already have the max fuel tank size equipped, but can go up to an A2 FSD that maxes out at 0.9T for a (longer) max distance jump.
You can't upgrade the fuel tank size, but you can add fuel tanks in "optional" slots, and I think everything comes with a configuration allowing at least a few jumps out of a single tank, even if you fully upgrade your FSD.
Note, however, that in-system flight also uses fuel, so you'll also use some getting to the station. In other words, you do not want to ever get too low on fuel, especially if you'll be doing a long in-system supercruise.
Instead of looking at the refueling amount when you're docked, you can look at the fuel gauge on the right side of your cockpit. That's the two horizontal bars under where it says "FUEL". The thicker fuel gauge bar is your main tank, used for frame shift jumps. When you have a jump location selected, part of the bar will be a different color, to show you how much fuel that jump will use. The thin bar on top of that is your "reserve" used for in-system flight. Reserve is replenished from the main tank when it runs out.
Also note that your max jump range will be higher if you're not carrying cargo, if your fuel tank isn't full, or anything else that reduces your total mass.
It's not clear to me how the fuel per jump works, but it does seem like a single jump at max distance uses more fuel than 2 jumps at half that distance. If you're using the Galaxy Map Navigation to get a route, I believe the "Economical" option gives you the lowest fuel usage and the "Fastest" gives you the least jumps.