What New Testament verse (I think it was St. Paul) says that we must first evangelize those who are already in the Church first before making converts of others outside the Church?
You may be thinking of Paul in his letter to the Romans:
For I am not ashamed of the gospel, for it is the power of God for salvation to everyone who believes, to the Jew first and also to the Greek.
9 There will be tribulation and distress for every human being who does evil, the Jew first and also the Greek, 10 but glory and honor and peace for everyone who does good, the Jew first and also the Greek.
Paul, who considered himself (confirmed by the other apostles) to be Apostle to the Gentiles, might have received the evangelizing priority of the Jews from Jesus himself. This is in line with some gospel passages set before Pentecost, where Jesus and his disciples deliberately focused the announcement of the Kingdom only to the Jews, while making it clear that eventually the Gentiles will be included, as it's obvious from passages in Isaiah.
Relevant passages in Matthew about the priority of the Jews:
Instruction to disciples in Matt 10:5-6:
5 These twelve Jesus sent out, instructing them, “Go nowhere among the Gentiles and enter no town of the Samaritans, 6 but go rather to the lost sheep of the house of Israel.
What Jesus said to the Canaanite woman from the district of Tyre and Sidon in Matt 15:24:
[Jesus] answered, “I was sent only to the lost sheep of the house of Israel.”
Matthew is written after Pentecost, several passages about evangelizing the Gentiles:
16 “Behold, I am sending you out as sheep in the midst of wolves, so be wise as serpents and innocent as doves. 17 Beware of men, for they will deliver you over to courts and flog you in their synagogues, 18 and you will be dragged before governors and kings for my sake, to bear witness before them and the Gentiles.
15 Jesus, aware of this, withdrew from there. And many followed him, and he healed them all 16 and ordered them not to make him known. 17 This was to fulfill what was spoken by the prophet Isaiah:
18 “Behold, my servant whom I have chosen, my beloved with whom my soul is well pleased. I will put my Spirit upon him, and he will proclaim justice to the Gentiles. 19 He will not quarrel or cry aloud, nor will anyone hear his voice in the streets; 20 a bruised reed he will not break, and a smoldering wick he will not quench, until he brings justice to victory; 21 and in his name the Gentiles will hope.”
John Piper wrote a thorough study on Rom 1:16, listing 6 ways on how the Jews have priority over the Gentiles, 3 ways on how the Jews don't have priority, as well as why Paul mentioned the topic in his letter. From his website article To the Jew First, and Also to the Greek, the 6 ways that Jews have priority are:
They are the historic chosen people of God.
They are the guardians of God’s special revelation, the Old Testament Scriptures.
The Messiah and Savior, Jesus, comes to the world as a Jew to Jews.
Salvation is from the Jews, since everyone who is saved is saved by being connected to the covenant with Abraham by faith.
The Jews are to be evangelized first when the gospel penetrates a new region.
The Jews will enter first into final judgment and final blessing.
For more explanation on Jesus's ministry to the House of Israel first, see the gotquestions.org website article Why did Jesus say that He was sent only to the lost sheep of Israel (Matthew 15:24)?
I was thinking of Galatians 6:10:
Therefore, whilst we have time, let us work good to all men, but especially to those who are of the household of the faith.
St. Paul shows the proper order of charity, that those closer to us deserve our charity more.
But because we cannot do good to everyone, he presents the order in which it is to be done, when he adds: but especially to those who are of the household of the faith, who, namely, are not only akin to us in nature but united by faith and grace: “You are no more strangers and foreigners: but you are fellow citizens of the saints and the domestics of God” (Eph 2:19). Therefore mercy must be extended to everyone but preferably to the just who share in the faith, because it is said: “But if any man have not care of his own and especially those of his house, he hath denied the faith and is worse than an infidel” (1 Tim 5:8).
But here it might be asked whether it is lawful to love one more than another. To answer this, it should be noted that love can be called greater or less in two ways. In one way, from the standpoint of the object; in another, from the intensity of the act. For to love someone is to will good to him. Accordingly, one can love one person more than another, either because he wills him a greater good, which is the object of love, or because he more intensely wills him a good, i.e., with a more intense love. Therefore, with respect to the first, we ought to love everyone equally, because we ought to wish the good of eternal life to everyone; but with respect to the second, it is not necessary that we love all equally, because since the intensity of an act results from the principle of the action, and the principle of the action is union and similarity, we ought to love in a higher degree and more intensely those who are more like us and more closely united to us. [cf. Summa Theologica II-II q. 26 on the order of charity]