The complete reservation can be be made 11 months (or 7 months for non-home resort) from the date of your check-out. If you are checking out on April 1 you are correct that May 1 would be the first day you can call. BUT, this year May 1 is a Saturday. MS is closed Saturday, Sunday and some holidays. In that case, you can call the first following business day -- Monday, May 3 in your case.
If you want to be absolutely certain of getting what you want on the first shot -- not usually necessary except for holiday times or for special rooms like BWV standard/boardwalk view or GVs, you are correct that you would call each day. You would call on March 28 to book the 27th, the 29th to book the 28th, etc. MS would then link them all together. You could call the first day and check availability and get MS guidance as to the necessity of calling day-by-day.
Good luck!