An editor at Britannica pointed out an artful use of ‘not’ when splitting an infinitive:
Be aware that putting “not” or another adverb between “to” and its verb adds some emphasis to that adverb. For example, in the sentence “They decided not to stay another night” the phrase “they decided” is the most important information, but the sentence “They decided to not stay another night” tells us that maybe they decided to stay another night before, but now it is important that they will not stay.
An excellent use of ‘not’, this is. Often, however, there are more expressive, positive alternatives to a negative splitting of the infinitive:
Jack hoped to not need another surgery.
Jack hoped to avoid another surgery.
Jack hoped to forego surgery by taking a holistic approach.
Jack hoped that proper rest and exercise would make additional surgery unnecessary.
It is best to pair ‘hope’ with some sort of positive action, not ‘not’. After all, Jack had plenty of work to do on the farm. He had no time for negativity.