I do batches of about 60 or so, and as soon as they start hatching I fire up the brooder to make sure its nice and HOT in there (95-100) Because of this same issue!!
I have actually had quails that broke the shells of those not even pipped, and kill the baby on accident (?) by pipping the shell fromt he exterior and hitting a major vein or the yolk!
Also, do NOT use shavings if at all possible for the brooder...paper towels, regular washcloths/cleaning cloths or the shelf liner work best since they do not stick to the sticky bodies and are easily thrown away or cleaned.
sprinkling some feed and having a VERY shallow water dish (thisnk soda botle cap-sized, or tupperware/butter tub lid) full of water in there for humidity and drinking. It will evaporate quickly with the high temp, so refill it often.
As soon as they are walking steadily you can put them in the brooder. MAke sure the temp is between 95 and 100 on the floor before you move them.
the most important part is keeping the weak ones in the hatcher, and take the strong ones away and put in the brooder. they DO kill and further weaken theose hatching and newly hatched!
I hope this helps you out!