Well Grant was not Lincoln, so I'm not sure if that's relevant.
You're right that they wouldn't fall neatly into modern categorization, but keeping in mind that the left-right dichotomy changes over time then Lincoln would almost certainly be considered left-wing.
Broadly speaking, the left seeks to reform the culture, and the right seeks to preserve the culture (or return culture to a previous status quo.) Lincoln was definitely a "reform the culture" sort of guy.








