This nifty secure baby pad is called a topponcino, and it is a Montessori staple for infants after they progress from the Moses basket. After the first three months in the Moses basket, your infant will move to the thin mattress on the floor. At this point, he or she needs only a topponcino, a light cotton pad (shown above) on top of the sheet.
Your infant's room should be kept at a comfortable temperature for his or her small body. Do not use pillows because they are too big and put your child at risk for suffocation.
If anyone makes topponcinos, please do post link in a comment here! We are always looking for good ones to recommend.