Here are some tips for taking similarly excellent newborn pics before you leave the hospital with your next little one.
(1) Use your bed sheet to give you a plain white background. That gives you good contrast, isn't distracting, and lets you white-balance off it in your editing software (anything from Picassa to Photoshop).
(2) Set up your "studio" right next to the window. This will give you a huge, soft, directional, natural light source. It would be best if you can shoot at a time of day when the sun is angling into your window, but mid-day is good. The pics below were taken around noon.
(3) Turn on every light in the room. This will provide you with as much secondary light as possible. If you want a higher level of contrast between light and dark areas, then turn all the lights off instead.
(4) If your camera will let you, set your aperture as wide as it will go. (My camera wouldn't let me.) Some point-and-shoot cameras have an aperture priority mode. Any DSLR will have one.
(5) Set your ISO as high as you can without making your pictures unbearably noisy. Many cameras will claim to have huge ISO ranges, but the top two settings are useless most of the time. Typically 800 is pretty safe; the pics below were shot at ISO 800. If you can't manually adjust your ISO, then you should put on you big girl panties and go buy a new camera.
(6) Turn your flash off.
(7) Be sure you are focusing on your baby's eyes. (This doesn't apply if his/her face isn't in the picture. :P)
(8) Get low and close and take some pictures. Don't get so close that your camera can't focus.
(9) Step back and zoom in on some pictures. You will get more separation of the subject (baby) and the background by backing off and zooming in.
(10) Don't wait for a smile. Babies make all sorts of cute and amazing faces. Get your newborn yawning, crying, looking at you, looking away from you, fighting with clothes, and whatever else. You can delete pictures later, but you can't recreate the moments you missed. That's a basic rule of thumb for digital photography; when in doubt, shoot it.
We had been cleared to be discharged when we were ready to leave, and this little D-I-Y session was a bit more hurried than I would have liked. Still, we got some pretty good pics without involving either my serious gear or the hospital photographer. Personally, I would have still let the hospital photographer take her pictures, but the two times she stopped by were bad times. As it is, I'm glad we took the time we did to get the pics we did.
I hope the info above helps you get better pics, too.