Mazal tov on your recent marriage!!
It is common for remnants of the hymen to remain after the first time a couple has relations. However, this does not always cause further bleeding or discomfort during intercourse. You should not look for any bleeding after relations. Wait a few minutes after completing relations before wiping yourself (and your husband) with tissues or colored towels, and don't look! If you go to the bathroom afterwards, do not look at the toilet paper. Make sure to have relations on colored sheets. You would only be rendered niddah if you experience an actual flow of blood or if your husband finds blood on himself.
We sometimes suggest that any further discomfort or bleeding from the hymen can be reduced by stretching the remainder of the hymen. This could be done in the bath by gently inserting a finger into the vaginal canal, moving in all directions to stretch the hymen in all directions. It sounds as if you are already able to do this. Furthermore, if you use a lubricant such as KY Jelly the next few times you have relations, you are also less likely to have bleeding. If despite all this advice (and not looking) you still have a problem, please feel free to get back to us.
B'Hatzlacha!