Amavasai tharpanam can be performed starting with Sapindikaranam and no need to wait till the Abdikam (annual ceremony).
After the conclusion of Sapindikaranam, whichever tharpanam occurs first, whether it be amavasai or masa pirappu tharpanam, one must start with that and one must mandatorily perform all tharpanams. Some perform masa pirappu tharpanam for 4 months only while other might perform all the 12 months.