I'm guessing that's a freewheel and not a cassette back there ? Anywho what do you mean "better range" as that's a wide range of gears and all of them quite useful. If you need something bigger than 48/14 you could probably coast just as well. The jump from 44/28 to the 26 front will be a big jump but you shouldn't be changing gears under load anyway. The four tooth difference in front makes each shift in between each change in back.

I think whatever "better" you'll get is better shifting with shimano cogs,derailleur and rings but the two close chain rings should be very quick and smooth. The problem is 'better' will be a whole new drive train from rings to wheel to derailleur, etc.

I think you can get away with a 32 cog with that suntour derailleur but you'll find out. The challange is finding a freewheel,,maybe someone has a 7 speed Sachs(?) freewheel out there and you can get the equivalent gears plus one bigger,,but do you need that low of a gear?

It's a good bike. I forget are the wheels 700c or 27"?

one other thought, if you find a 7spd freewheel it might have a 13t cog so you'll end up with some larger jumps in the middle with the higher and lower gears.