At first, I was just going to comment on your comment on the other answer, but maybe a different approach is better.
We can look at this two ways. If we build it up, it goes like this:
- 出る - to come out
- 出ない - to not come out
- 出なく - a "stem" form conjugation of 出ない (Japanese verbs and adjectives share some conjugation rules) such that it can be put in front of another verb. [not really sure how it would translate it English maybe "not coming out" ?]
- 出なくなる - to start to not come out. (possibly in the future). Here we are adding the helping verb なる (= to become)
- 出なくなって - it is starting to not come out or it has started to not come out [continuative form of naru to express a state of not coming out]
So 出なくなって is the て-form of the helping verb なる when it's helping 出ない [the negative form of 出る]
出なくて means something slightly different. It means "it's not coming out." and is the て-form of 出ない
The difference being that 出なくなって indicates a change has occurred where it was coming out and now is not coming out whereas 出なくて indicates merely that it is not coming out.
So if tried to milk a cow and nothing ever came out, 出なくて but if you milked a cow and it stopped providing milk, then 出なくなって.
In other words, 出なくなって is not a verb sequence. It's a verb plus a helping verb. If you're looking for it in Tae Kim, then it's here.