He should ram a voting rights act through Congress. With the Great Migration not having occurred yet, if all the black voters in the South can all vote GOP, the Democrats will be screwed.
Not going to happen.
Roosevelt is
on record (in a private letter circa 1914) as stating that actual full democracy couldn't work in the South because of the large number (majority in many areas and two states) of blacks - whom he considered incapable of responsible citizenship.
It was one thing for modest numbers of blacks to vote in areas where they were small minorities and couldn't cause too much damage. But he agreed privately with the contention of white-supremacist Southerners that blacks in power would be shamelessly corrupt and also incompetent.
The vestigial Republican Party organizations in the Deep South had gone entirely "Lily White" by this time, and when Roosevelt wanted their votes at the national convention, he assisted in their exclusion of blacks. (Ironically, there were fairly strong Republican parties in the Upper South at this time. Tennessee elected a Republican governor in 1910, 1912, and 1920.)
By that time, most northern Republicans agreed - again privately. That is why nothing was done under Roosevelt or in the 1920s to stop lynching, or enfranchise southern blacks.