This response comes a bit late, sorry for that. As described here by u/Antiquarianism historians are very sceptical of Moses as a historical figure, so for that reason it is difficult to find other prophets in his time, but there were definitely those who claimed to be prophets in the times of Jesus (at least the period between the founding of Judaea Province and the Jewish Revolt). However, they did not produce texts as far as we know, and they were either killed or forced into hiding by the Roman leadership.

The Judaeo-Roman author Josephus mentions several miracle-workers and rebel leaders who can be compared to Jesus of whom some claimed prophethood, though those all did it after Jesus' lifetime. Josephus writes (Jewish Antiquities 20.97-98) that between 44 and 46 AD, a man named Theudas claimed to be a prophet and gathered a large crowd with the promise of dividing the River Jordan (like Moses or Joshua in the Old Testament), until the governor Fadus sent Roman cavalry which killed or captured many of his followers and executed Theudas himself (The Acts of the Apostles in the Bible (5:36) also has a Pharisee elder mention Theudas in a comparison with Jesus, but this has caused a lot of confusion since it is claimed that Theudas rose up before Judas the Galilean, who revolted against Roman taxation in the year 6 AD). There was also an anonymous leader often called simply "the Egyptian Prophet" mentioned in both Josephus' Jewish Antiquities (20.169-171) and Jewish War (2.259-263), who claimed prophethood and gathered followers on the Mount of Olives where he promised to tear down the walls of Jerusalem and take over the city. Like with Theudas the Roman governor, at that time Antonius Felix according to Antiquities, sent soldiers to disperse his followers, but he himself managed to escape and did not appear again. Josephus mentions two more similar figures who he does not explicitly call false prophets but who are presumed to be: the "Samaritan prophet" (Antiquities 18.85-87) who gathered armed followers on Mount Gezirim with the claim that he would find objects buried by Moses until Pilate sent troops, and the "Anonymous prophet" (Antiquities 20.188) whose followers wanted to go with him into the desert, before, once again, he and his followers were killed by Roman forces.

Now all of these appeared after Jesus' death (though The Samaritan just a few years afterwards), but there had also been some people before who were similar to him, holy men and miracle-workers, though they did not claim prophethood nor were suppressed by the Romans. Josephus tells us (Antiquities 14.21-28) that a few decades before Jesus there had been a man "beloved of God", Onias, who had stopped a drought by praying for rain. The Talmud, using his Hebrew name Honi records the same story in more detail, with him drawing a circle and refusing to step out until God makes it rain. Also in the Talmud it is said that he was in a magical sleep for decades, which some commentators have seen as being similar to Jesus' death and resurrection (though in Josephus' account he is in fact stoned to death for refusing to take sides in a conflict between two Judaean priest-kings).

There was also John the Baptist, discussed in both Josephus' Antiquities (18.5.2) and The Bible. Though the Gospels are very insistent that he saw Jesus as greater than himself and as the Messiah it is difficult to know what he himself claimed. But even the Gospel of Mark compares him to Elijah and states that many believed Jesus was a resurrection of John. Josephus also states that many Jews thought King Herod's army was destroyed as God's punishment for killing John, which could indicate that he was seen as more than a regular person. He is also today seen as a prophet in both Islam and Mandaeism.

So there were definitely people who claimed to be prophets in the decades after Jesus' lifetime, and people who were seen as miracle-workers and who gathered followers before him. Unfortunately we have no writing about them other than Josephus and (for some) the New Testament, and most were killed by authorities.