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On January 27th, 1796, George Washington nominated Samuel Chase to replace John Blair and William Cushing to Replace Rutledge as Chief Justice. Cushing declined the nomination and Washington nominated Oliver Ellsworth in his stead on March 3rd. Both Chase and Ellsworth were confirmed on the day after their appointments. Washington was not running for reelection.
John Adams nominated John Jay to replace Ellsworth as Chief Justice on December 18th, 1800. John Jay declined the nomination so Adams nominated John Marshall to the post on January 20th, 1801. Marshall was confirmed by a voice vote of the Senate a week later. Five weeks after that, on March 4, Marshall swore in Thomas Jefferson as President. This appointment was part of a effort by the Federalists to creat a court system which ran counter to the ideals of Thomas Jefferson and the soon to be dominant Democratic-Republican Party.
According to Wikipedia:
The lame-duck session of Congress in late 1800 enacted the Judiciary Act of 1801, which created a set of federal appeals courts between the district courts and the Supreme Court. The purpose of the statute was twofold—first, to remedy the defects in the federal judicial system inherent in the Judiciary Act of 1789, and second, to enable the defeated Federalists to staff the new judicial offices with loyal Federalists in the face of the party's defeat in 1800–the party had lost control of both houses of congress in addition to the White House.[98] Adams filled the vacancies created in this statute by appointing a series of judges, whom his opponents called the "Midnight Judges" because most of them were nominated just days before his presidential term expired. Most of these judges lost their posts when the Jeffersonian Republicans enacted the Judiciary Act of 1802, abolishing the courts created by the Judiciary Act of 1801 and returning the federal courts to their original structure as specified in the 1789 statute.
Jefferson nominated William Johnson to replace Moore on March 22, 1804. He was confirmed by voice vote two days later. At the time the Democratic-Republican Party dominated the Congress.
On December 17, 1828 John Quency Adams nominated John J. Crittenden to replace Trimble. Andrew Jackson's supporters in the Senate delayed the vote and on February 12, 1829 the Senate voted 23–17 to postpone further proceedings on the appointment. Jackson removed Crittenden from his post as district attorney when he came into office and two days later John McLean was confirmed on March .
Andrew Jackson initially nominated Roger B Taney to serve as an Associate Justice to replace Duval on January 15th, 1835 but the nomination was postponed on March 3rd by a vote of 24–21. Subsequent to the death of Chief Justice John Marshall on July 6th Jackson nominated Taney again on December 28th to be Chief Justice. At the same time Jackson nominated Philip Pendleton Barbour to replace Duvall. Both were confirmed on March 15th, 1836.
On his last day in Office President Jackson nominated John Catron and William Smith to two newly created seats. The incoming Congress confirmed them both on March 8th but the pro slavery Smith refused to serve.
After the December 18th, 1843 death of Justice Smith Thompson and the April 21st, 1844 death of Justice Henry Baldwin, John Tyler went through his final year in office trying to replace the two justices. Tyler nominated John Canfield Spencer to replace Thompson on January 9th, 1844 but that nomination was rejected on a 21–26 vote on January 31st. Tyler subsequently nominated New York Chancellor Reuben Walworth for the post on March 13th but his name was withdrawn on June 17th. Tyler again nominated Walworth after the withdrawal but the Senate took no action on the second try. Tyler nominated John Canfield Spencer for the post on June 17th but the nomination was withdrawn the same day. Tyler tried again with Walworth on December 4th, 1844 but the nomination was withdrawn on February 4th, 1845. Tyler was finally able to seat Samuel Nelson in Thompson's stead with a February 4th nomination which was confirmed 10 days later.
President John Tyler was unable to replace Henry Baldwin. He nominated Edward King forte seat on June 5, 1844 but the nomination was postponed with a 29–18 Senate vote on June 15. King was again nominated on December 4. King's second nomination was withdrawn with the February 7, 1845 nomination of John M. Read. No action was taken by the Senate on the Read nomination and incoming President James Polk finally replaced Balwin with a August 1846 nomination of Robert Cooper Grier. Polk failed in his first attempt to replace Baldwin with his December 1845 nomination of George Woodward was rejected by the Senate.
James Filmore had similar luck in his attempt to replace John McKinley after hid death on July 19th 1852. Filmore nominated Edward A. Bradford on August 16th but no action was taken. On January 3rd, 1853 George Edmund Badger was nominated by Filmore but the nomination was withdrawn on February with the nomination of William C. Micou. No action was taken on the Micou nomination and President Franklin Pierce successfully appointed John Archibald to fill McKInley's seat on March 21st .
President Lincoln nominated the anti slavery Salmon P. Chase to replace the deceased, pro slavery, Taney as Chief Justice December 6th, 1864 after he had already won the 1964 election. Chase was confirmed that same day.
After the 1972 election Ulysses S. Grant nominated Ward Hunt to replace Nelson on December 3rd, 1872. He was confirmed the next week.
After William Strong's retirement in December of 1880, President Hayes nominated William Burnham Woods. He was confirmed the next week.
On January 24th, 1881 Noah Henry Swayne's retired on the condition that President Hayes replace him with Stanley Matthews but the Senate took no action on the January 26th, 1881 nomination. Incoming President Garfield successfully renominated Mathews shortly after he took office.
President Grover Cleveland successfully nominated two Justices during the year leading up to his 1988 defeat by Benjamin Harrison. The Senate had the majority in the Senate while Cleveland was a Democrat. Lucius Quintus Cincinnatus Lamar replaced Woods a month after his December 1887 nomination and Melville Fuller replaced Chief Justice Waite with a July 1888 confirmation vote.
During President Taft's final year in office the Senate confirmed Mahlon Pitney to replace Harlan on March 13th, 1912.
Woodrow Wilson had a friendlier Democratic Congress when he placed two justices on the court while facing reelection. Louis Brandeis was confirmed on June 1st, 1916 to replace Lamar four months after his nomination while John Hessin Clarke replaced Hughes the next month only 10 days after his nomination.
Herbert Hoover had a friendly Republican led Senate which confirmed Benjamin N. Cardozo nine days after his nomination of February 15th, 1932.
The Democratic Super-majority took 12 days to confirm Roosevelt's nomination of Frank Murphy in January of 1940.
Finally we have Johnson's failed nominations to replace Earl Warren and Reagan's nomination of Anthony Kennedy. Both of those are covered in recent diaries.
Most of the ones I mention are also covered. Hopefully I have expanded the scope somewhat to include a few obscure individuals and nominations which show that Supreme Court Nominees, especially during a President's final years, have resulted in History which shaped where we are today.