We use arcs found by Storme and van Maldeghem in their classification of primitive arcs in PG(2,q) as seeds for constructing small complete arcs in these planes. Our complete arcs are obtained by taking the union of such a ``seed arc'' with some orbits of a subgroup of its stabilizer. Using this approach we construct five different complete 15-arcs fixed by Z3 in PG(2,37), a complete 20-arc fixed by S3 in PG(2,61), and two different complete 22-arcs fixed by D5 in PG(2,71). In all three cases, the size of complete arcs constructed in this paper is strictly smaller than the size of the smallest complete arcs (in the respective plane) known so far.