Right, if OP has this one then they could keep the 3x shifter and replace the 7x one, if they go 8/9x. But the easiest/cheapest way to lower gears is a smaller chainring. And the easiest way to bigger range is a 7x cassette with bigger range, if it exists.

I found a SRAM/Sachs Spectro 3x7 manual at https://sheldonbrown.com/sram/dtm_ghs_my00_e.pdf
According to this, the hub is 135 mm OLD, which means newer 8/9/10 speed versions of dualdrive or Sturmey Archer 3x cassette hubs will fit the frame, in case other solutions don't work.

It also says that a 12-28t cassette is standard (and that others may cause malfunction, but I wouldn't worry much about that on a small wheel bike). And it is indeed a cassette hub, not threaded for freewheels. I've seen that 7-speed 13-34t cassettes exist, this would also give lower gears in a simple way, while sacrificing a top gear.