QUESTION: Does an evangelist evangelize those already converted, or those yet unreached with the gospel?

ANSWER: Both! Timothy as an evangelist (II Timothy 4:2) was told by Paul to instruct the sinner (those who oppose themselves) that they might acknowledge the truth and recover from the snares of the devil (II Timothy 2:25-26; I Timothy 6:17). He also was told to teach the brethren (I Timothy 4:6).