higher up vs to a higher place

higher up and to a higher place both are adverbs.

Word	Noun	Adjective	Verb	Adverb
higher up	No	No	No	Yes
to a higher place	No	No	No	Yes

As adverbs, higher up and to a higher place are synonyms defined as:

Other synonyms of higher up include above, in a higher place.

higher up (adverb)	to a higher place (adverb)
in or to a place that is higher	in or to a place that is higher