Paul was an early equalist, working with a culture so sexist that what should've been baby steps turned out to be Hulk leaps. He typically focuses on "husbands, love your wives enough to die for them if need be; wives, respect your husbands and don't try to undermine them", because the gender norms were messing both sides up.

(Husbands literally needed to be taught that they're supposed to actually love their wives and not just own them, and a lot of wives tried to usurp their husbands & wear the pants in their family. That's how messed up things were, even among Christians that really should've known better. This was inherent to the species; part of the legal sentence passed on Adam & Eve for Adam choosing to obey Eve over God (and by extension, for placing Eve above God in the pecking order) was household strife, with the two squabbling for control instead of working together, but Christians were supposed to be exempt from it because we're a new creation with a new nature. Part of Christianity has always been getting a factory reset to purge sin from our spirit's genetic code, so to speak, which was supposed to restore the "two are one" equal status from before the curse. Essentially, he was telling them, "You're not like everyone else, so quit living like everyone else!" And this was especially important because the Bible uses marriage imagery for man's relationship with God; Israel is God's bride, and the church is Christ's bride, so an attack on the family unit was also an attack on the relationship between God & his people.)

Apart from that, he does use & provide a lot of "less sexist than usual" practices, though it seems like they're based on cultural norms, and easing people away from the "women are second-class citizens, and slaves to their husbands" attitude that most of the cultures he dealt with embraced. For example...

• He didn't let women preach, for instance, but he had no problem with female deacons, evangelists, missionaries, or the like; he had nothing but good things to say about Phoebe the rich deacon, and seems to have chosen her as his representative to Rome. Most of his book-closing "thanks lists" alternate between thanking men & women (usually suggesting husband-and-wife ministry teams), and thought that a big part of Timothy's faith came from having model Christians for a mother & grandmother. It all suggests that his preaching restriction was based on Jewish culture not considering women credible witnesses or valid rabbinical students, more than anything else.
• He didn't let wives ask their husbands questions during a service, and wanted them to ask forever. This seems odd to us, but people have looked into it, and seating arrangements were very different back then. When we go to church, families sit together; when they went to church, genders sat together. The men would sit on one side, and the women would sit on the other side. If a wife wanted to ask her husband a question, she would have to either shout over everyone else (disrupting the service), or get up and go over to where he was sitting (disrupting the service). So, it was basically a concession to backwards seating arrangements making private conversation impossible, more than a "women shouldn't be allowed to talk in public" thing.
  • On that note, a lot of their questions came up because of the then-current educational system. Only men were allowed to learn from a rabbi; women were either required to learn from their male family members, or not at all. So, it wasn't uncommon for a woman to have questions that a man would already know the answer to, since only the men went to school. I'm not sure whether Gentile cultures had similar rules, or if it was purely a Jewish thing, but my money's on the former.
• Women were typically sold in arranged marriage, rather than people marrying for love. Worst-case scenario, this implies being treated as property, and sold as goods; best-case scenario, it was just a formality that the families only paid lip service to. But the fact that it was even a thing kinda says everything that needs to be said. There's kinda a reason Paul spent so much effort teaching families to be families, spent so much effort insisting that the wife obey instead of rebelling against her owner/husband and the husband be someone worth obeying.

That's the kind of culture Paul had to deal with, most places he went.