The trouble is (at the start anyway), very little is private. You rarely get time to yourself and even then, it's not private.
I really have no good ideas to help. It depends a lot on your specific circumstances - where basic training is, what type of advanced training you will take, your rank, where you will be station, etc. All of these can change how to cope with the situation.
The only idea I can come up with is to get a small plushie, as small as possible that can help you . That way you may be able to keep it hidden better and maybe pass it off as a present from your 'girl friend' or a family member.