Romana's the only one whose ever been shown to influence how her regeneration looked, and nothing about that regeneration made any sense anyway.
We don't know how much of an influence a Time Lord can have on their future regenerations. We have seen two cases where a Time Lord had some control. Romana was shown to have great control over her regeneration. She changed her appearance four times before settling on her original choice of "Princess Astrid" (Though this regeneration was mostly done for comedic purposes). The Master, after being shot by Chan-Tho, wanted to be "young and strong" like the Doctor. He regenerated from an old man to a younger and stronger man.
Also, out of universe, it depends on who is casted for the role. If the actor isn't ginger, the Doctor won't be ginger, although the doctor jokes he wants to be.
Also the Sisterhood of Karn can influence regeneration with their Elixer of Life. See my hyperlinks for more information.
(different person) Time Lords have the ability to focus and control what their regenerations will be like, like how Romana and the Master did. River Song also focused on a dress size, and she got it. The only reason the Doctor doesn't focus on being a ginger is because he is usually dying a rather horrable and painful dead to focus on being a ginger.
The most likely way is that the Doctor would have gotten desperate enough to waste a regenerationnby injesting an elixer from the Sisterhood of Karn to make himself ginger.
However, if he could hold off on regenerating like the Tenth Doctor, then he COULD INDEED could to Karn and ask for an elixer to make his next incarantion ginger.