It is clear from the Shulchan Aruch and the acharonim that the above 2 options are the proper times to say selichos. What about other times?
The Mateh Efraim discusses saying selichos after Alos and says that b'dieved this is fine and that it is better to say selichos after Alos then not say them at all.
The Acahronim discuss one other time, the first half of the night. Most Acharonim are very much against this as the first half of the night is a time of din and in fact, R' Ovadya Yosef states that it is better not to say selcihos at all then say them in the first half of the night.
R' Willig told us that based on this, that instead of saying selichos at 10:00PM, amuch better choice is to daven mincha a little early and say them between mincha and maariv at the end of the day before sunset.
To sum up there are 4 possible times to say selichos
1. Early morning before Alos - best time to say selichos
2. Right after Chatzos - also l'chatchela
3. During the day - anytime after Alos until sunset, not l'chatchela but better to say selichos at this time then not say them
4. First half of the night - according to some Acharonim it is better not to say selichos at this time, according to others this is really b'dieved (if you absolutely have no other choice).
The above is only a general guideline, as always, ask your local Rav any specific questions.
Here are some specific references about the issue
Mishnah Berurah 565:12, selichos should not be recited before midnight.
Sha'arei Teshuvah 581:1 quoting Birkei Yosef, one who finds himself in a shul where selichos are being recited before midnight, should not recite the Thirteen Attributes along with the congregation.
Igros Moshe O.C. 2:105, R' Moshe is very against the practice of saying selichos before midnight.
Yechave Da'as 1:46, prohibits saying selichos before midnight, instead he advises reciting selichos before Minchah.