I've been told that it's correct with this example:
遅れちゃった、寝過ごしたので。
But I don't understand why it's used like this because ので is the conjunctive form of のです so it's used to connect the sentence with the result after (I know that the result can be omitted) but it's not anymore logic if the result is given before, why not simply use のです in this case ?
「遅{おく}れちゃった、寝過{ねす}ごしたので。」
is a completely normal sentence in the real Japanese-speaking world. Japanese word order, phrase order and clause order are far more flexible than it appears to me that they are taught in Japanese-as-a-foreign-language.
Thus, it is an everyday occurence for certain conjunctions to be located at the very end of the sentences. You will often see/hear a sentence structure such as:
「(Result/Effect Clause)、+ (Reason/Cause Clause) + ので、から, etc. 。」
which is virtually the same in meaning as:
「(Reason/Cause Clause) + ので、から, etc. + (Result/Effect Clause)。」
Conjunctions in general are often placed at the very end of the sentences, which seems to baffle some learners. Those conjunctions would include:
が、けど、けれども、のに、なのに, etc.
I believe this a case of a so called "dislocation" which happens in speech (or a writing that imitates speech) - grammar of a sentence (寝過ごしたので遅れちゃった) is broken in the way that the latter part is said first (it's what's on the speaker's mind) and the rest is added for more information/clarification, often as an afterthought ... resulting in the sentence you quoted.
So yes, in a normal grammatical sentence the ので would be attached after a reason that would be stated first, followed by a consequence (either explicit or just unstated, implied).
