…And not a new convert, so that he will not become conceited and fall into the condemnation incurred by the devil.
1 Timothy 3:6 (emphasis mine)

Later in this same letter Paul emphasizes the importance of being patient when considering men for church leadership.

Do not lay hands upon anyone too hastily and thereby share responsibility for the sins of others; keep yourself free from sin.
1 Timothy 5:22

Spiritual maturity is an essential quality for biblical leadership. The Scriptures often warn us about the professing Christian who is not growing properly in the faith.

And I, brethren, could not speak to you as to spiritual men, but as to men of flesh, as to infants in Christ.
1 Corinthians 3:1

For though by this time you ought to be teachers, you have need again for someone to teach you the elementary principles of the oracles of God, and you have come to need milk and not solid food. For everyone who partakes only of milk is not accustomed to the word of righteousness, for he is an infant.
Hebrews 5:12–13

Paul warns us that a new convert can easily become “conceited.” He uses this word conceited in one other place in 1 Timothy and again in 2 Timothy. Notice the context.

If anyone advocates a different doctrine and does not agree with sound words, those of our Lord Jesus Christ, and with the doctrine conforming to godliness, he is conceited and understands nothing; but he has a morbid interest in controversial questions and disputes about words, out of which arise envy, strife, abusive language, evil suspicions…
1 Timothy 6:3–4

But realize this, that in the last days difficult times will come. For men will be lovers of self, lovers of money, boastful, arrogant, revilers, disobedient to parents, ungrateful, unholy, unloving, irreconcilable, malicious gossips, without self-control, brutal, haters of good, treacherous, reckless, conceited, lovers of pleasure rather than lovers of God…
2 Timothy 3:1–4
(emphases mine)

He uses this word intentionally when addressing the qualifications for eldership. The idea is that when a man becomes exceedingly proud, he also becomes foolish. The result can be disastrous. We know that the devil fell because of pride. He will certainly exploit this sinful pattern in others and will cause untold damage to the man and the church.