Not only does grammar like order, it likes balance. And that first sentence is unbalanced. Just as either needs or, and neither needs nor, not only is part of a matched set and needs its other half. Usually, that’s but also.
All those expressions are “correlative conjunctions,” and they “must frame syntactically identical sentence parts,” Garner’s Modern American Usage says. In other words, both sides of the sentence being joined should be parallel.
Despite what Mrs. Whatnot may have told you in English class, but also is not your only option for following not only. You could say any of these things and still be balanced:
Not only does grammar like order, it likes balance as well.
Not only does grammar like order, but it likes balance too.
Not only does grammar like order, it also likes balance.
The needed requirement is something that says “and.” After all, the basis of the phrase is “Grammar likes order and balance.”
Not only needs a partner; but also can hang by itself: Grammar likes order, but it also likes balance.
Of course, not only is this parallelism often abused, not everyone agrees it’s needed. Merriam-Webster’s Dictionary of English Usage says: “So long as you take care that the groups of words joined by the conjunctions are not so dissimilar as to call attention to themselves, you need not worry all the time about achieving precise parallelism. It is more important for your sentence to sound natural and to make sense.”
Not only is that true for parallelism, it’s good advice for all of English. As well.