What were the population demographics in the South during the Hayes/Tilden election?

Your question relates to the changing balance of power between Republicans and Democrats between the 1868 and 1876 Presidential elections, particularly in the southern United States. Your particular concern is the representation of African Americans in 1876, and whether possible suppression of their votes "unfairly" deprived Hayes of a popular vote majority to support his electoral college majority.

It is noteworthy that the total suffrage rose from 5.7 million in 1868 to 6.4 million in 1872 8.3 million in 1876. (This is for the two major parties and excludes "splinter" candidates' votes.) During this time, the Republican vote rose from 3.0 million to 4.1 million, or 1.1 million, while the Democratic vote rose from 2.7 million to 4.2 million, or 1.5 million.

Most of the difference in the Democratic vote rise came in the southern states. The issue was that in 1868, many whites, were unable to vote, either because their individual states had not been reconstructed, or because the voters themselves had not taken the loyalty oaths or whatever was needed to establish themselves as voters. By 1876, whites had fully regained their suffrage.

I'm going to use a simplifying assumption: That most southern white voters voted Democratic, and that most black voters voted Republican, unless otherwise specified. So looking at only the southern states, I divide them into three groups. 1) states where the percentage of the Democratic voters exceeded the percentage of whites (the states of concern), 2) states where the percentage of the Democratic vote in 1876 approximated the percentage of whites, and 3) "special cases" where the percentage of the Democratic vote in 1876 was much less than the percentage of whites were much less than the percentage of Democratic voters.

In South Carolina, for example, the white population in 1870 was "only" 41%, but the Democrats won almost 50% of the vote in 1876 (versus 24% in 1872). This represented a huge surge and disproportion in the Democratic vote, reflecting, among other things, a 101% turnout of eligible voters, as pointed out by T.E.D. Such "ballot-stuffing" also suggests activity of the opposite kind, suppression of the "wrong" votes.

The remaining states that show similar patterns, are presented as State (whites as % of population, 1876 Democratic percentage, 1872 Democratic percentage). Similar patterns existed in Georgia (54%, 72%,54%); Alabama (52%, 60%, 47%), and Mississippi (46%, 68%, 47%). These states' 1872 Democratic vote percentage were more or less in line with their percentage of whites, the disparity occurred in 1876;

Of less concern are states in the second category, They are: Virginia (56%, 60%,59%); Louisiana (50%, 48%, 44%), and Texas (69%,70%. 59%).

In the last group are states where whites voted Republican in significant numbers, meaning that the Democratic share of the vote was much less than the white population. North Carolina (63%, 54%, 42%), Arkansas (76%, 66%, 48%), Tennessee (77%, 60%,52%), and Kentucky (82%, 61%,52%).

Sources: 1870 census. Wikipedia articles for Presidential elections in 1868, 1872, and 1876.

I've been looking at answering this for days, but in the end I think the question itself can't be properly answered for a couple of reasons.

The first is that we didn't have exit polling back then, so we don't really know what rate white people voted at, or even what rate black people voted at. Of course even then, some amount of Southern white voters were voting for Republicans, and some (tiny?) amount of black voters would vote for Democrats.

The second is that Reconstruction was still going on, so Jim Crow was not entirely set up yet. In fact, merely by looking at the county map, and comparing it to the Black Belt map, we can see which states were giving African Americans something like their full say, and which weren't. The difference between the Louisiana side of the Mississippi River and the Mississippi side is particularly stark.

It seems pretty clear just from a glance at these maps that Mississippi and Georgia weren't letting their black citizens vote, Louisiana, Arkansas, and Florida mostly were. South Carolina, North Carolina, Virginia, and Alabama were in lots of places, but were suspiciously spotty.

In point of fact, at this time Louisiana, South Carolina, and Florida still had Federal troops in them protecting the rights of black citizens.

The third is that there was in fact shenanigans to an extent that it was simply not what any reasonable person could call a fair democratic election, so even if we did know those rates, simply applying them selectively wouldn't fix things. For instance, in South Carolina, birthplace of the Confederacy, 101% of voters voted in this election. That's a turnout record that hopefully will never be surpassed! You certainly can't fix the count there by speculatively adding voters to it.