Sounds like you're describing rev-matching. That's different from double clutching. Double-clutching is when you engage the clutch momentarily with the gearbox in neutral, to get the input shaft spinning at the right speed. This saves wear on the synchro. (Unless you fumble it...)

Rev-matching is tapping the gas pedal as you engage the clutch after you have downshifted. This is what prevents the jerk. This doesn't save your synchros, but it prevents the shock-load on your driveshafts, clutch, gearbox, & engine mounts.

You can double-clutch to shift gears, then engage the clutch without matching revs; that still jerks. OTOH, you can shift without double-clutching, then match revs when you engage the clutch, & that's smooth.

Rev-matching is always good. Double clutching is 'usually' not needed in normal cars, normal driving. But its still fun & sounds cool if you don't screw it up.