The most common and most likely answer to this crossword clue is the 6 letter word NOBLER
I've seen this crossword clue in Independent
N
O
B
L
E
R
More clues leading to the result NOBLER
Buddy explains!

The straight part
We know that "more high-minded" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case more high-minded = nobler.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.
What does NOBLER mean?
NOBLER adjective- Having honorable qualities; having moral eminence and freedom from anything petty, mean or dubious in conduct and character.
- Grand; stately; magnificent; splendid.
- Of exalted rank; of or relating to the nobility; distinguished from the masses by birth, station, or title; highborn.
- (of a polyhedron) Both isohedral and isogonal.