The Democrat ran to the right of the Republican in at least 1904, 1912, 1924, 1928, 1932, and 1960, and arguably in 1952,* 1956,* and 1976, and a case could be made for 1992.
*depending on how much weight one assigns to civil rights
The parties have only aligned into coherent entities quite recently, but the position of the Democrats as the party of (what would now be considered) economic leftism wasn't solidified until the New Deal.
I see where your coming from as far as the Democratic Party of the early 1900's through (with some exceptions) the 1950's.