The two usual changes up are one semitone and one tone. The tone is probably the better one, as it uses two notes from the original tonic in the new dominant, whereas the semitone uses one. Either has been used successfully. Going up (?) to G is the next in the circle of fifths, but may well be difficult to sing, as the range will be much higher (or lower).
Don't know why this was voted as best answer. Maybe it answered the un-edited question. For the new answer, it will depend entirely on the tessitura of the vocalist that can't reach certain notes. As before, a tone, tone and a half, is usual, but that will also impact on the ability of the accompanying musician to play in the new key. C is fine on the guitar, Eb not so for some. Same on keyboards. A capo is good for taking keys up, not so for going down. I sing a particular song with one guitarist, who refuses to take it down a tone to suit my range, because of the new chords.