Hagan is quite mistaken. Let me elaborate still further.
Josephus says in Antiquities 20.10.1 that, “Accordingly, the number of the high priests, from the days of Herod until the day when Titus took the temple and the City, and burnt them, were in all twenty-eight”
Emil Schurer provides this: “a list of Pontiffs of Herodian-Roman times is given here. Josephus says that they numbered twenty-eight, and the following twenty-eight names are arrived at by collating his various accounts” (History of the Jewish People in the Age of Jesus Christ, II:228). There is no place in this list of 28 high priests for Jonathan in between Ananius and Ismael.
What is more, notice that Josephus always gives details on the deposition and
Besides which, he also deprived Joseph, who was also called Caiaphas, of the high priesthood, and appointed Jonathan the son of Ananus, the former high priest, to succeed him. (18.4.3)
Herod the tetrarch and his friends, went up to Jerusalem to offer sacrifice to God, an ancient festival of the Jews being then just approaching; and when he had been there, and been honorably entertained by the multitude of the Jews, he made a stay there for three days, within which time he deprived Jonathan of the high priesthood, and gave it to his brother Theophilus. (18.5.3)
And when Agrippa had entirely finished all the duties of the Divine worship, he removed Theophilus, the son of Ananus, from the high priesthood, and bestowed that honor of his on Simon the son of Boethus, whose name was also Cantheras (19.6.2)
And now king Agrippa took the [high] priesthood away from Simon Cantheras, and put Jonathan, the son of Ananus, into it again, and owned that he was more worthy of that dignity than the other. But this was not a thing acceptable to him, to recover that his former dignity. So he refused it, and said, “O king! I rejoice in the honor that thou hast for me, and take it kindly that thou wouldst give me such a dignity of thy own inclinations, although God hath judged that I am not at all worthy of the high priesthood. I am satisfied with having once put on the sacred garments; for I then put them on after a more holy manner than I should now receive them again. But if thou desirest that a person more worthy than myself should have this honorable employment, give me leave to name thee such a one. I have a brother that is pure from all sin against God, and of all offenses against thyself; I recommend him to thee, as one that is fit for this dignity.” So the king was pleased with these words of his, and passed by Jonathan, and, according to his brother’s desire, bestowed the high priesthood upon Matthias. (19.6.4)
And now he took the high priesthood away from Matthias, and made Elioneus, the son of Cantheras, high priest in his stead. (19.8.1)
Accordingly, Herod removed the last high priest, called Cimtheras, and bestowed that dignity on his successor Joseph, the son of Cantos. (20.1.3)
But now Herod, king of Chalcis, removed Joseph, the son of Camydus, from the high priesthood, and made Ananias, the son of Nebedeu, his successor. (20.5.2)
But then when he mentions Jonathan, after the appointment of Ananias, he says nothing of Ananias being deposed or of Jonathan being appointed high priest, nor does he tell us what Jonathan he’s talking about (Jonathan son of ???). Why? Because he has already introduced the famous Jonathan before. We already know who this is. It’s not a new appointment, it’s the ex-high priest.
Felix also bore an ill-will to Jonathan, the high priest, because he frequently gave him admonitions about governing the Jewish affairs better than he did, lest he should himself have complaints made of him by the multitude, since he it was who had desired Caesar to send him as procurator of Judea. So Felix contrived a method whereby he might get rid of him, (20.8.5)
Josephus then resumes this pattern of detail on deposition and appointment:
About this time king Agrippa gave the high priesthood to Ismael, who was the son of Fabi. (20.8.8).
As soon as the king heard this news, he gave the high priesthood to Joseph, who was called Cabi, the son of Simon, formerly high priest. (20.8.11)
But the king deprived Joseph of the high priesthood, and bestowed the succession to that dignity on the son of Ananus, who was also himself called Ananus. (20.9.1)
king Agrippa took the high priesthood from him, when he had ruled but three months, and made Jesus, the son of Damneus, high priest. (20.9.1)
But as for the high priest, Ananias (25) he increased in glory every day, and this to a great degree, and had obtained the favor and esteem of the citizens in a signal manner; for he was a great hoarder up of money: he therefore cultivated the friendship of Albinus, and of the high priest [Jesus], by making them presents . . . So the other high priests acted in the like manner, . . (20.9.2)
Notice that they are still called “high priest” even though they do not hold the office any more.
And now Jesus, the son of Gamaliel, became the successor of Jesus, the son of Damneus, in the high priesthood, which the king had taken from the other; on which account a sedition arose between the high priests (20.9.4)
Again, the ex-high priests are still called “high priest”.
He also deprived Jesus, the son of Gamaliel, of the high priesthood, and gave it to Matthias, the son of Theophilus (20.9.7)
Steve, the record is clear. There were twenty eight high priests in this period according to Josephus himself. There is no place for a high priest between Ananias and Ismael. By Josephus’ own testimony Ismael was appointed right about the time Festus took over from Felix, c. AD 59. Ananias was the high priest when St. Paul was in Jerusalem, just as St. Luke says. Josephus and St. Luke agree. Hagan has gotten this wrong.