New upper bounds on the smallest size t2(2,q) of a complete arc in the projective plane PG(2,q) are obtained for 853≤q≤5107 and q∈T1∪T2, where T1={173,181,193,229,243,257,271,277,293,343,373,409,443,449,457,461,463,467,479,487,491,499,529,563,569,571,577,587,593,599,601,607,613,617,619,631,641,661,673,677,683,691,709}, and T2={5119,5147,5153,5209,5231,5237,5261,5279,5281,5303,5347,5641,5843,6011,8192}...