Hi Ron, unfortunately I have no grammatic answer to your question. There are subtle differences between these words you gave and I will try my best to explain.
1. 新：is often used as the antonym of 旧（used, refer to things）, or 老（known, refer to people）: for example: 新房子，旧家具，新同学，老朋友；when it is used with the meaning of 另一个, in most cases it implies that there is a used / existing one (旧 / 老) to give a feeling of making contrast, for example: 我需要一个新老师 (I am not happy with the teacher I have now.) It is similar to 别的，the difference is elaborated in #3.
- Let's make a new appointment: 我们约个新的时间吧。
2. 再：if you use 再, in most cases it means that you are not setting up a time right now, but are saying this in order not to appear rude (that you don't care if this appointment is made or not, while in reality you may or may not really care). It actually means "see you around" - 再约吧。再约时间吧。我们再约吧。我们再约时间吧。
3. 别的：is similar to 新, only it is used as the antonym as 现有的 （existing, refer to both people and things）。
- Let's make a new appointment: 我们约个别的时间吧。约个别的时间吧。
4. 另一个：is more specific than 别的 or 新 and is not used in making a general statement, such as "let's make a new appointment"。when you use 另一个，it normally means that you were given two options, and you have to change the time to the other. For example, you were given 2pm and 4pm as options, you went with 2pm, but something came up and you had to change the time. Considering the options you were given before, you thought 4pm would make better sense to try than other times (such 10am). So you said: 我们能约另一个时间（4pm）吗？
However, what you can say, is 我们另外约一个时间吧 to mean “let's make another appointment".
另外：can ben seen as an adjective, meaning to start over, in spite of what's said or done, start anew.
5. another word you can use to say "let's make another appointment" is 其他，it is similar to 别的 - Let's make a new appointment: 我们约其他时间吧。
Hopefully I've made this clear. Hope this helps. Cheers. Happy learning!