Q: Were sacrifices still offered in the 48 levitical cities on the local altars even after Solomon's temple was completed, so folks could offer a sacrifice more than three times a year when they went to Jerusalem for annual feasts? I understand no other sacrifices were to be offered in Jerusalem after the First Temple's completion, and God no longer allowed the high places to be used for sacrifices to Him after that, but I can't find anywhere if those living far away from Jerusalem simply could no longer sacrifice except three times per year. I know orthodox Jews don't sacrifice still to this day because they have no Temple in Jerusalem.
A: To answer your question, no, it was not permitted to offer sacrifices outside of the Temple once the Temple was erected in Jerusalem and that is where people had to go. What we do know is many Jewish people disobeyed that rule because they did not want to make the journey and so sacrificed at what the Bible calls “the high places”. These were smaller versions of the Temple and had a Holy of Holies and Holy Place and so on, and one such high place has been discovered in the City of Arad in the Negev, but there were several of these throughout the country. While many Jewish people chose to bring the sacrifice there because of convenience, these were not acceptable with God.
Because the sacrifices were offered to the true God but in the wrong place, is one reason why the many good Kings of Israel would not do away with them as they did away with idolatry. Only exceptionally good kings actually did away with the high places as well.
The 48 Levitical cities were simply places for the Levites to live in since they received no land possession for themselves as a tribe. So those who were willing to obey the Lord had to bring the sacrifices to the Temple in Jerusalem and this was normally done three times a year during the three pilgrimage festivals.
As far as individual sacrifices were concerned, these were performed as necessary but the three annual festivals had to be observed only in Jerusalem. They were not allowed to sacrifice in the Levitical cities per se but only where the Temple stood.