Yes, you are dead right that now the low end needle(they called it 'LSN' - low speed needle) has to be tuned also. It's a good idea to change the plug as well, as it might be coated with some dirt during the running-in process.
Do not be afraid on tuning the low end, otherwise you would have unstable idling & frustrating engine stallings due to flooding. I would usually go at 1/4 turn each from the start as the factory's setting is always very rich. What you do is, fire up the engine and warm it up well first. Next, tune the HSN(high speed needle) till you get good power & a light trail of smoke at full throttle. At this stage of your engine rather new life, do not run at maximum full power but slightly on the rich side to prolong it's lifespan. After that's done, bring the bug in to a full stop beside you. Count 5 secs(like... 1 thousand 1, 1 thousand 2~5), then open up the throttle fully. At this stage, the bug would definitely bog(or hesistate) before it starts to gather power after it has clear out the excess fuel loaded up during the 5 secs count due to the current rich setting. Bring it in, turn the LSN clockwise(lean down) 1/4 turn & try the same 5 secs count respond again. Do that until it has reach the stage where it responds instantaneously & shoot off like a rocket cleanly.
Take note here that when you notice it start to respond better & better, tune by 1/8 increment instead of 1/4 turn. As you keep leaning down, the idling speed will naturally start to go up due to leaner setting now. It is therefore necessary to re-adjust by backing out the idling stop screw anti-clockwise to get a stable idling speed.
Once you have completed the above, the engine should fire up on the first bump next time and always. It would also be able to idle for one full tank(highly not
) without stalling again. A new engine is the easiest to start if it is properly tuned. That's the most wonderful part about changing to a new engine, but there's a trade off with the most boring part of the running-in process.