Need more context to really give you the right answer. In the context provided, it means to moved side to side to the top of the hill. With more context, it actually describes that the animal simply moved up the hill, taking its time and pleasantly. Not making any sharp turns or in a rush.

The context of the sentence is useful, but the context of the situation might be also. Whatever it is, it seems quite interesting. 'To sway' means to move slowly from side to side. 'To sway off' kind of suggests something swayed ungently and moved away from where it was before. 'To sway off and up' means that something moved away in such a manner that resulted in it arriving on top of the hill.