First of all, congratulations!! Since you mentioned that you already have a baby I'll assume it's doing well? If I have found a baby so soon after a birth I take note of the level of aggression between the adults. It can get pretty nasty and I always am concerned that the baby may accidentally get bitten by a frenzy of protective aggression towards any invasion when the cage is entered. Of course I've tried to not handle the baby much and at least during the first examination of it, you may want to use vinyl gloves for sanitation as well as not letting any foreign scent get on it. I try to measure total length (TL), Snout to vent length (SVL)in cm and weight in grams. I have found that most of babies born here are from 90 to 100 grams. As the next few weeks pass they seem to thrive off of the 'yolk'(?) sac in their cute round belly. A second weight taken during the next few weeks may actually be lower than the original weight so just look at the animal itself to evaluate its progress. I've assumed this drop in weight is normal, having seen this result occur with other animals. It's also very important to examine the umbilicus to make sure it's closing/healing properly. I may dab it with a q-tip with iodine or povidone solution automatically at least once, then again in a few days if it still looks open. I've been lucky not to have any babies with problems with healing there. At this time I also try not to have the humidity too high or the bottom too wet for fear of fungal growth in the cage.
Thus far,I have found that the most critical issue at this time is to make sure the baby stays hydrated. I seem to have had little problems with this by making sure that there are multiple feeding bowls AND that one of them is very shallow (I use a plastic pot saucer for this), offering food in it WITH A LITTLE BIT OF WATER AS WELL. I have this hope that the skinks can learn to drink from a water bowl and I think I've seen it work with adults too over time. There may even be an advantage where they may not poop in that bowl when you also provide a water bowl that's big enough for them to lay and poop in.
Finally, during the next few weeks I make sure to visually check the baby for injuries from bites from the adults. Sometimes it happens, especially if you are trying to catch the baby or put it back but I have not yet had to remove a baby from parents. Again, I think I only try to handle the baby twice during the next few months to avoid this problem. It's hard to resist though. Oh yeah, if you can, photo document its pattern as a baby because it changes a little as they grow.
If I can think of other tips on babies I'll pass them along.
Take care and enjoy your Xmas!
Darweedy