The NL East-leading Mets also claimed right-hander Roel Ramírez off waivers from the St. Louis Cardinals.
The Mets acquired McKinney from Milwaukee on May 25. In 39 games with New York, the 26-year-old batted .220 (20 for 91) with six doubles, a triple, five home runs and 14 RBI. He was designated for assignment on July 16.
The 23-year-old Rincon hit .263 with 12 homers and 48 RBI in 63 games at Double-A Tulsa. He will report to Double-A Binghamton.
The 26-year-old Ramirez will be optioned to Triple-A Syracuse. He pitched in one game for St. Louis this year and one game last season.
Ramirez was 0-1 with one save and a 4.34 ERA in 20 games with Triple-A Memphis this year.