I think there is a 9 month halacha on getting remarried after a previous marriage. Perhaps I am mistaken.
You are mistaken. The halacha is three months. The reason for the halacha is to be able to establish paternity of any potential children. Once three months have passed, it is usually clear if the divorcee/widow is pregnant or not and hence she can remarry*.
It should be noted that it’s not clear (to me, anyway) if this only applies to marriage, or if it applies to dating as well. In other words, can a woman who has been divorced/widowed for two months begin dating (with the understanding that she will not get married until after the three month period has passed)?
As for the question of the OP, there are no set rules. She should begin dating when it feels right to do so (subject to possible halachic restrictions as mentioned above). It’s really based on individual preference.
The Wolf
(*And yes, the halacha applies even in situations where it is well-known that the woman cannot be pregnant by her husband — age, her husband was away, etc.)