Did he though? His efforts achieved exactly the same thing for both of them - 2/3rds approval for both amendments in congress.
But only two states ratified the first amendment. If President Lincoln was such a strong supporter of that amendment, as you claim he was, you would think that he would have pushed harder for ratification. On the other hand, passage of the second 13th Amendment was added to the Republican platform with President Lincoln's enthusiastic support and he lobbied for passage of the amendment out of Congress and saw it ratified by 21 states before he was murdered.