A single size of tube is able to accommodate many sizes of tire. Sometimes the tube is folded and sometimes it is stretched inside the tire. As long as the tube is not pinched between the rim and sidewall of the tire then there will not be a problem.
Try to figure out why your old tire was having problems. Sometimes it is caused by a brake pad hitting the tire and cutting the threads. If this was the cause of the first tire failure, it is likely to be the cause of a second.
Also, an out-of-true rim is much less likely to be felt in the saddle than a bulge in a tire.