The best way, when it works
It usually works to grab the whole mass of coils, and flip them over when handing them to the second. When you're doing this, care has to be taken that you don't wrap one of the ends around one of the climber's anchor cord. The leader (when bringing up the second) has to make sure that the loops are neatly and tightly stacked, and not falling over.
Also, as you're coiling: The general advice is to start with large loops, and progress to smaller ones as the second gets closer to the belay. That way the loops don't interfere with each other as you belay from that coil. With this setup, you want to do the opposite - start with smaller / medium sized loops, and progress to larger ones. That way when you flip it, the larger loops are on the bottom.
But this whole thing is more art than science, and I'd recommend you try and practice it safely on the ground (maybe in your living room) a few times to get the hang of it. If the loops aren't laying neatly, this technique probably won't work, and will result in a big knot. Also, you have to make sure that the person leading the next pitch (or their end of the rope) isn't somehow tangled as you pass the coils over.
The slower, more predictable option
You also have the option of recoiling the rope manually, passing it foot-by-foot from the leader to the belayer. This is going to be slower, but has a lower chance of getting tangled. If you think the lap coils aren't as neat as you'd like, or your belayer is inexperienced, and may have trouble managing the coils while belaying, I'd go with this route.