candidates supported by both major parties
As early as the first Federal elections in 1788, there were candidates, who while alligned with one party or another, was supported in the press by both parties in a particular election.
Elector: A member of the Electoral College chosen by the several States to elect the President and Vice-President.
Oxford English Dictionary
See President of the United States for the votes cast by the Electoral College for President.
Historical Note: These elections were vastly different from modern day Presidential elections. The actual Presidential candidates were rarely mentioned on tickets and voters were voting for particular electors who were pledged to a particular candidate. There was sometimes confusion as to who the particular elector was actually pledged to. Prior to the 12th Amendment, electors were pledged to two candidates as there was no distinction made in the Electoral College between President and Vice-President.
1789 - 1824: Alabama, Connecticut, Delaware, Georgia, Illinois, Indiana, Kentucky, Louisiana, Maine, Maryland, Massachusetts, Mississippi, Missouri, New Hampshire, New Jersey, New York, North Carolina, Ohio, Pennsylvania, Rhode Island, South Carolina, Tennessee, Vermont, Virginia
State Breakdown for Election of Members to the Electoral College:
Alabama: Legislature (1820), Popular Vote (1824)
Connecticut: Legislature (1789-1816), Popular Vote (1820, 1824)
Delaware: Popular Vote (1789), Legislature (1796-1824)
Georgia: Legislature (1789, 1792, 1800-1824), Popular Vote (1796)
Illinois: Popular Vote by District
Indiana: Legislature (1816, 1820), Popular Vote by District (1824)
Kentucky: Popular Vote by District
Maine: Popular Vote by District
Maryland: Popular Vote (1789, 1792), Popular Vote by District (1796-1824)
Massachusetts: Popular Vote (1789-1796, 1804, 1824), Legislature (1800, 1808, 1816), Popular Vote by District (1812, 1820)
Mississippi: Popular Vote
Missouri: Legislature (1820), Popular Vote by District (1824)
New Hampshire: Popular Vote (1789-1796, 1804-1824), Legislature (1800)
New Jersey: Legislature (1789-1800, 1812), Popular Vote (1804, 1808, 1816-1824)
New York: Legislature
North Carolina: Legislature (1792, 1812), Popular Vote by District (1796-1808), Popular Vote (1816-1824)
Ohio: Popular Vote
Pennsylvania: Popular Vote (1789-1796, 1804-1824), Legislature (1800)
Rhode Island: Legislature (1792, 1796), Popular Vote (1800-1824)
South Carolina: Legislature
Tennessee: Popular Vote by District
Virginia: Popular Vote by District (1789-1796), Popular Vote (1800-1824)
Office Scope: Federal
Role Scope: State / District