Incidentally, the House was severely hung at the start of the 34th Congress in 1855-1856. Starting on December 3, 1855 it took 133 ballots before the House finally selected Nathanial P. Banks (A-MA) as Speaker on February 2, 1856.
(Note: the first regular session of Congress began in December thirteen months after the election and nine months after they took office back then.)
Congressional and presidential terms all began on March 4. To be sure that the House had time to elect a President if necessary, they arranged it so the outgoing Congress would count the electoral votes in January, and have a couple of months to elect a President before their term ended. This meant the presidential electors had to vote in December and be elected in November, which is why we hold elections on the 2nd Tuesday after the 1st Monday in November.
Since the second session was at the end of the term, they also delayed the start of the first session until December (which is the default set in the Constitution). The first session, or long session, would last into the spring or summer. For the 33rd Congress:
33rd (4 March 1853-3 March 1855)
Special SENATE 4 March 1853-11 April 1853
9 month gap at start of term.
1st 5 December 1853-7 August 1854 (8 months in session)
4 month gap
2nd 4 December 1854-3 March 1855 (3 months in session)
The Senate would hold a special session at the beginning of an incoming President's term in order to confirm his cabinet and any other appointees. 2/3 of the senators continued in office, others would have been re-elected by their legislatures, and at least some of the newly elected senators might make a short trip to be sworn in and introduced. This special session was proclaimed by outgoing President Millard Fillmore, to coincide with the start of President Franklin Pierce's term.
Congressional elections were not held on a uniform date. A very few States would hold elections on the same day as a presidential election, but most would hold them at the same time as their statewide and legislative elections, which were often set in the State constitutions. A fall election would permit legislators to begin a term on January 1 for a winter legislative session.
The Kansas-Nebraska Act was passed in 1854, signed by President Pierce on May 30, 1854, during the first session. This resulted in the collapse of the Whigs, and the organization of the Republican Party, and an opportunity for the American Party, for those who didn't want to be a Republican or Democrat. The first session continued through August.
About 3/5 of representatives were elected in the fall of 1854 as many in October as November, but with some in August, September, or December. The winners would be congratulated, and then have 4 or 5 months before the start of their term, but more than a year before they would actually sit in Congress or be sworn in. The incumbent representatives, including those who were retiring, or had been defeated would then go off to Washington for the second or short session.
About 2/5 of the representatives were elected after their term begin, mostly in the fall of 1855. This was particularly true in Southern states. After being elected in October or November, they would travel to Washington and be sworn in.
Neither the Republicans (108) or Democrats (83) had a majority. Eventually, Nathaniel Banks who had been elected on the American Party ticket was elected as Speaker.
Banks was re-elected to Congress as a Republican in November 1856, so would finish up his speaker-ship after being elected by a different party than the one he had been when elected speaker. As a result of the 1856-57 elections, the Democrats took control of the House, and soon after the 1st session of the 35th Congress began in December 1857, Banks resigned having been elected governor of Massachusetts.
