Second quesadilla made on my panini press is not as crispy as the first
Probably its a panini-press-not-heated-enough issue, but with a caveat. I dont know about the inner wokings of the press, but it sounds like this: the thermometer that decides if the light for "ready" sign goes green is situated in a place that gets heated early, gives green, and cuts off the heat production (while the rest of the press is was still gathering heat.) 

I am inclined to say that this is a not-hot-enough issues as, if you think about it, every new quesadilla you cook is actually a "second quesadilla", just with enough time lapse between the last quesadilla and this quesadilla.