The other thing that may affect the swap is how the 13T sprocket is designed. The 13T sprocket for use with a 11T is usually different from a 13T sprocket for a 12T.

A 11T sprocket usually has a thicker integrated spacer that 'nests' part way into the 2nd sprocket, and this will affect the spacing between the sprockets.

Notice in the pics the difference in the shape of the sprockets face in the area where the smallest sprocket would sit. The 13T sprocket for a 12T 2nd sprocket is flat. The 13T sprocket for a 11T sprocket has a recessed ring where the thicker spacer would sit.
13T for 12T sprocket

13T for 11T sprocket