My DS was born around the same gestation but was 5lb11oz so was a bit bigger then your DD and he stayed in his bassinet until he was 6 months. We moved him because he didnt like to be swaddled and his hands started hitting the side of the bassinette and was startling him during the night. So we moved him. We still had to have a towel rolled up under the matress.

Is there any reason in particular you want to move her to the cot?