I know that before the Civil War, there was concern about maintaining the balance of slave states and non-slave states. Did any of the states after the Civil War face significant political hurdles before they were admitted?
Hawaii's admission was delayed because of
"Of course, [Sen.] Lyndon Johnson was no friend of statehood ... There were 22 times when he voted against us ... He did everything he could, because he was representing the Southern racial opposition." - an interview with Betty Farrington (non-voting delegate to the House of Representatives 1953-1957)
The compromise: admit Alaska just before.
Sources: http://archives.starbulletin.com/1999/10/18/special/story4.html , http://history.house.gov/People/Detail/13007
Alaska had a referendum in 1946: the vote for statehood was just 9,630 to 6,822. A major concern locally was whether there were enough revenues to support a state.
In 1952, a bill died in the Senate under a threat of Southern flibuster. In the 1954 State of the Union address, Eisenhower said "The people of Hawaii are ready for statehood. I renew my request for this legislation in order that Hawaii may elect its State officials and its representatives in Washington along with the rest of the country this fall." Hawaii was then a Republican state; Alaska was a Democratic state. (Side note: in the next paragraph, Eisenhower asked for a constitutional amendment lowering the voting age to 18 because 18-year-olds were fighting for the US.) In 1954, "Frustrated by Eisenhower refusal to support statehood for Alaska, a Senate coalition led by Democrats ties the fate of Alaska and Hawaii Statehood together as one package. The parliamentary move is backed by some Southern Democrats, concerned about the addition of new votes in the civil rights for blacks movement, in the hope of defeating both measures."
Sources: https://www.alaska.edu/creatingalaska/downloads/Statehood-for-Alaska.pdf , http://ankn.uaf.edu/ANCR/Southeast/ChronologyofDates , http://www.vlib.us/amdocs/texts/dde1954.htm
"Still, numerous Democrats in the U.S. House spoke against admission of Wyoming--known to be leaning Republican. In the minority in the House, Democrats knew they’d be unable to block admission based on party affiliation alone--but they could argue against statehood for a Republican territory by attacking women’s suffrage. They made persistent objections to Wyoming's article 6 that granted women equal rights. It almost worked." http://www.wyohistory.org/essays/wyoming-statehood
South Dakota, North Dakota, Montana, and Washington were admitted in an omnibus bill in early 1889 in the lame-duck Congress. They were Republican-leaning areas; Republicans took the White House and Congress in 1888. Idaho and Wyoming were added in 1890. John E. Miller, "MORE THAN STATEHOOD ON THEIR MINDS: SOUTH DAKOTA JOINS THE UNION, 1889", http://digitalcommons.unl.edu/cgi/viewcontent.cgi?article=1393&context=greatplainsquarterly
It's getting really late here and I'm tired, so I hope I can be forgiven for just writing: Utah, Mormons, polygamy, Mormon political party.
"Congress and the Territorial Expansion of the United States" at http://www.princeton.edu/~nmccarty/state.pdf looks like it is nicely comprehensive and informative, but I've not read it in detail. It became noticeably partisan in the late 1800s, as mentioned above.
Just to clarify, are you asking about any states admitted into the Union after the Civil War, or the former Confederate states (and territories) re-admitted into the Union after the Civil War?
Although getting close to this sub's time limits one can also note that the voters in the federal district of Washington, DC called for a constitutional convention in 1980 and drafted a proposed state constitution in 1982. They have faced significant political hurdles in being admitted as the 51st state up to the time limit of this sub. For example in 1993 losing a vote in the House of Representatives by a vote of 277 to 153 and having further bills to grant statehood being killed in committee.