Peaceable Projects Inc. is a U.S.-based non-profit group that brings the vast resources of the wide world together with the ongoing needs of the people who live, work, and travel on the Camino de Santiago pilgrim trail network in Spain.
Are you asking in order to avoid large crowds? The Spaniards are mostly on the section from Sarria to Santiago, in the months of July and August. Is that the route section you are planning? In SJPP, there is a surge in pilgrims (largely non-Spanish) starting in September and early October. The "best time" will depend on your starting point and your wish for company.
As @ C clearly states: starts from SJPDP in September and early October is normally a surge. With the relaxation of COVID travel restrictions, in general and travel from North America, in particular, I believe the surge will be greater than usual. I, for one, depart SJPDP August 26th and finish in SDC, on or about, October 1. I always travel alone. Not because I'm anti-social, but rather my intent is to meet pilgrims I may never have known, the exception being those I've met on the Forum. I pretty much let the Camino guide me along the Way with few expectations other than finishing in one piece.