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I am saying that for the most part yes the believers were pretty much in Jerusalem until the persicutions began. This was the beginning of the spreading of The Gospel to further places on a broad scale.

Acts ch 11 19Meanwhile, the believers who had fled from Jerusalem during the persecution after Stephen’s death traveled as far as Phoenicia, Cyprus, and Antioch of Syria. They preached the Good News, but only to Jews. 20However, some of the believers who went to Antioch from Cyprus and Cyrene began preaching to Gentiles£ about the Lord Jesus. 21The power of the Lord was upon them, and large numbers of these Gentiles believed and turned to the Lord. 22When the church at Jerusalem heard what had happened, they sent Barnabas to Antioch. 23When he arrived and saw this proof of God’s favor, he was filled with joy, and he encouraged the believers to stay true to the Lord. 24Barnabas was a good man, full of the Holy Spirit and strong in faith. And large numbers of people were brought to the Lord.

Or KJV

Now they which were scattered abroad upon the persecution that arose about Stephen travelled as far as Phenice, and Cyprus, and Antioch, preaching the word to none but unto the Jews only.

Going back to Acts chapter 2

41Those who believed what Peter said were baptized and added to the church—about three thousand in all. 42They joined with the other believers and devoted themselves to the apostles’ teaching and fellowship, sharing in the Lord’s Supper and in prayer. The Believers Meet Together 43A deep sense of awe came over them all, and the apostles performed many miraculous signs and wonders. 44And all the believers met together constantly and shared everything they had. 45They sold their possessions and shared the proceeds with those in need. 46They worshiped together at the Temple each day, met in homes for the Lord’s Supper, and shared their meals with great joy and generosity—47all the while praising God and enjoying the goodwill of all the people. And each day the Lord added to their group those who were being saved.

Now KJV

Act 2:41 Then they that gladly received his word were baptized: and the same day there were added [unto them] about three thousand souls. And they continued stedfastly in the apostles' doctrine and fellowship, and in breaking of bread, and in prayers. And fear came upon every soul: and many wonders and signs were done by the apostles. And all that believed were together, and had all things common; And sold their possessions and goods, and parted them to all [men], as every man had need. And they, continuing daily with one accord in the temple, and breaking bread from house to house, did eat their meat with gladness and singleness of heart, Praising God, and having favour with all the people. And the Lord added to the church daily such as should be saved.

Yes I believe until the death of Stephen those added to the church at Pentecost remained in Jerusalem. The Persicutions which began that day drove the believeras out of Jerusalem and into neighboring cities and nations. Looking further The Disciples seemed to have their hands full taking care of them.

Acts chapter 6

1But as the believers rapidly multiplied, there were rumblings of discontent. Those who spoke Greek complained against those who spoke Hebrew, saying that their widows were being discriminated against in the daily distribution of food. 2So the Twelve called a meeting of all the believers. “We apostles should spend our time preaching and teaching the word of God, not administering a food program,” they said. 3“Now look around among yourselves, brothers, and select seven men who are well respected and are full of the Holy Spirit and wisdom. We will put them in charge of this business. 4Then we can spend our time in prayer and preaching and teaching the word.” 5This idea pleased the whole group, and they chose the following: Stephen (a man full of faith and the Holy Spirit), Philip, Procorus, Nicanor, Timon, Parmenas, and Nicolas of Antioch (a Gentile convert to the Jewish faith, who had now become a Christian). 6These seven were presented to the apostles, who prayed for them as they laid their hands on them. 7God’s message was preached in ever-widening circles. The number of believers greatly increased in Jerusalem, and many of the Jewish priests were converted, too. Stephen Is Arrested 8Stephen, a man full of God’s grace and power, performed amazing miracles and signs among the people. 9But one day some men from the Synagogue of Freed Slaves, as it was called, started to debate with him. They were Jews from Cyrene, Alexandria, Cilicia, and the province of Asia. 10None of them was able to stand against the wisdom and Spirit by which Stephen spoke.

Read the issues the disciples faced. I think the verses pretty well say where the believers were staying. At Stephens death the believers scattered and because they scattered The Gospel spread to further places. This era from Pentecost to Stephen's death was short. I've looked at several time lines and it could have been two years or as many as 4-6.

Paul's conversion by some sources was 35 AD. Meaning by that time was fairly short from the Pentecost to persicution.