Taking a bath at the sacred Kaveri is considered special during the month of Aippasi/Tula maasam. Taking a bath on the last day of that month is not considered that important and since one has to wait until the following year Tula maasam, though not special, at least one has to take a dip on the last day of that month.
A lame person who could not walk managed to reach for a bath on the first day of Karthigai month, yet he was granted the benefits of Tula Kaveri snanam per the story. The purpose behind this is one will gain the benefit of Tula snanam even if he takes a bath on the last day of Aippasi or on the first day of Karthigai month.