Aside from the hints mentioned in the question about Kōichi having returned to the town one and a half years ago but not having a memory of it, there are other hints most of which appeared earlier chronologically but are purposely left to the end in the anime series.
It's revealed afterwards that the extra desk that Countermeasures tried to account for and prevented in the classroom at the beginning
was actually in the Teacher's Lounge,
and previous incidents seemed to imply the Extra was always a student attending the class, so no one thought to check there.
Near the final revelation on who the Extra was, we see the continuation of a conversation Reiko had with Kōichi earlier in the anime which was cut off when it was initially shown,
where Reiko reminded Kōichi that she was the assistant teacher/deputy homeroom teacher of Kōichi's class and told Kōichi to refer to her by her proper name Ms. Mikami (Mikami-sensei) at school.
Along with the first hint, we see her connection with class 3-3.
We learn that Kōichi's mother, Ritsuko, was in class 3-3 with Misaki Yomiyama, the first Extra which sparked the Calamity. We also learn that
Reiko was in class 3-3 in 1983 when Ritsuko, being Reiko's sister, was a victim of the Calamity, and Reiko was class 3-3's homeroom teacher in 1996 when she died. We also know that victims of the Calamity can also be teachers, given how the current Class 3-3's homeroom teacher Mr. Kubodera is listed as one of the "Deaths of July".
Source: Characters - Yomiyama North Middle School - Faculty
There is one hint however which can easily be missed throughout the series and this is in regards to the Mikami Family Pet, Rei.
Rei is effectively a parrot who repeats phrases said around him, the 3 most common ones are
- "Good morning Rei!"
- "Poor Reiko, poor Ritsuko"
- "Why, Rei, why?"
We later learn that Rei was purchased after Reiko's death and thus 2 lines match up to that of grieving in the house. the 3rd line is probably the grieving questioning of Reiko's death and the point to it (as she was stabbed which lead to her drowning). the 2nd line however can be a hint that Reiko is the Extra due to the fact it is a repeated line showing sympathy to Reiko in the same manner as Ritsuko who was already dead and there doesn't seem to be a relative to them who died beforehand who Reiko and Ritsuko may have been devastated by.
This next part is speculation but whenever Rei starts talking Reiko begins to have headaches
since Rei is just parroting lines and probably doesn't understand what it is saying the Calamity's Memory Altering effects are probably not applied to Rei whoever Reiko's headaches may be her starting to become self-aware she is the Extra by understanding Rei's lines however weather by an effect by the Calamity or by her nature as the Extra Reiko headaches flare up and she seems to want to sleep them off thus stopping the process
The Calamity is designed so that it's near impossible to discover who the Extra was until after everything is over, as people's memories and official records are altered. The only difference that the current year had over all the previous years was that Mei Misaki had the power to see the Colour of Death.
During the anime, she demonstrated this power with the photos where Extra was known. Mei could see the colour of death around the Extra, and as such she could identify the Extra, which she did at the very end to confirm to Kōichi.
Oddly enough, however, Mei had never encountered the Extra given the Extra's position in the cast, with the exceptions to when the students went insane at Sakitani Kinenkan and prepared to kill Mei (Mei saw the Extra before the final confrontation however given the deteriorating situation she would've been unable to confront the Extra or speak to Kōichi). Also, the Countermeasures was never made aware of Mei's power even after she was allowed to exist.
There is also another aberration. Calamity normally alters memories; however, Kōichi's grandfather's memory is not altered due to him suffering from dementia.
He is the only character who lives in Yomiyama but still remember [the Extra's] death. This is due to his dementia conflicting with his Curse-altered memories.
Source: Ryouhei Mikami - Trivia
Due to the fact that
Ritsuko and Reiko sound similar, Kōichi could have passed it off as Ryouhei's dementia making him confuse between Ritsuko and Reiko.
For those who believed in the Calamity, particularly those in Countermeasures, this might have been a clue. However, since Kōichi had only just enrolled, the Countermeasures didn't have proper time to interview and prepare him, and thus Kōichi didn't learn about the Calamity until much later. Furthermore, the Countermeasures had already prepared the correct number of seats and had Mei play the role of "the one that does not exist" well before Kōichi's enrollment into the school.