No. You only need to pay the amount equivalent to the remaining days until the next billing, which occurs the 1st of each month.
For instance, let's say you pay USD9.99 for each player on your current subscription. If you have subscribed to one more player license the 15th, you only need to pay for the next 15 days of the month. This will total about USD 4,99. The following month you will be fully charged for that additional player.