"See you next week."
for reference, また when used as prefix with temporal nouns as in また来週, it changes the temporal noun as in "again next week".
while if you use また after a temporal noun, it would probably mean you are changing another thing, maybe a verb as in 「来週、また行きましょう」"next week, let's go again"
if you are still wondering this, it's not correct, you can use that order if you are planning on saying to do something next week like「 来週また会いしましょう」"let's meet again next week".
In the case of また来週 and またね, these are both phrases that mean "again, next week/ok?". I know is confusing but the reason why is a different use of また is because in this sentence「来週また会いしましょう」 the また is changing the verb, while the 来週 is changing the whole sentence through the verb, again, you can visualize it like this 来週「また→会いしましょう」
In the case of また来週, the また is changing the noun like また→来週, this is also an expression like saying おはよう it's is just how Japanese people use it.