The six are among the 13 states with "trigger laws" designed to ban or severely restrict abortions once the Supreme Court overturned the landmark 1973 Roe v Wade ruling that recognized a right to the procedure, as it did on Friday.
In Louisiana, abortion services that had been halted since Friday began resuming after Orleans Parish Civil District Court Judge Robin Giarrusso on Monday issued a temporary restraining order blocking the state from carrying out its ban.
The order came shortly after Hope Medical Group for Women in Shreveport - one of Louisiana's three abortion clinics - sued, arguing Louisiana's trigger laws "lack constitutionally required safeguards to prevent arbitrary enforcement".
Later on Monday in Utah, 3rd District Court Judge Andrew Stone, at the request of a Planned Parenthood affiliate, issued a temporary restraining order that would allow abortion services to resume in the state after a 2020 ban took effect on Friday.
"Today is a win, but it is only the first step in what will undoubtedly be a long and difficult fight," Karrie Galloway, president and CEO of Planned Parenthood Association of Utah, said in a statement.
Republican Louisiana Attorney General Jeff Landry said his office was "fully prepared to defend these laws in our state courts, just as we have in our federal courts." Republican Utah Attorney General Sean Reyes did not respond to requests for comment.
In Republican-led Texas, where a ban on abortions after six weeks of pregnancy went into effect last year, a judge in Harris County will hear arguments on Tuesday on whether to block officials from enforcing pre-Roe v Wade abortion prohibitions.
Republican Attorney General Ken Paxton had said in a Friday advisory that while the state's 2021 trigger ban would not take effect for 30 days after the Supreme Court's ruling, prosecutors could immediately pursue cases based on pre-1973 laws.
In Idaho, a Planned Parenthood affiliate asked the state's highest court to block the enforcement of a "trigger" law banning abortion that the Republican-controlled state legislature passed in 2020 due to take effect Aug 19.
Similar lawsuits were filed by abortion providers asking state courts to block the enforcement of "trigger" bans in Kentucky and Mississippi.
"To be clear, there is no right to abortion contained in the Commonwealth's Constitution - and we will stand up against any baseless claim to the contrary," Kentucky Attorney General Daniel Cameron said in a statement.
Abortion rights advocates plan to challenge an Ohio ban on abortions after six weeks that took effect on Friday, and in Florida, a group of abortion providers went before a judge on Monday to argue a challenge to that state's ban on abortions after 15 weeks of pregnancy.
In states where federal court orders were blocking abortion restrictions based on the Roe precedent, those orders are now being lifted. South Carolina's ban on abortions after a fetal heartbeat is detected is now in effect, Attorney General Alan Wilson said on Monday, after a federal judge put on hold an injunction that had prevented its implementation.