I was just thinking we all make fun of Donald Trump for the fake tan and bad hair. But I wonder what the reaction from the public would be if he just one day decided, for example, he was going to embrace male pattern baldness and ditched the combover. Has anything like that ever happened before in the past? Or is a president’s steady appearance a visual indication of their steady policies (or some baloney like that)?
I don't know if it exactly addresses your question head on, but Abraham Lincoln dramatically changed his appearance by growing a beard, probably shortly after he was elected on 6 Nov 1860, but prior to his inauguration on 4 March 1861.
Here is a photo on file at the library of Congress dated to 13 August 1860. Contrast that with this photo, identified by the Library of Congress as the earliest surviving photograph of the 16th president with a beard, dated to ~13 Jan 1861, about two months before his inauguration. Here he is again in February, just before leaving for Washington, showing a slightly fuller beard.
It's popularly reported that Lincoln's beard was inspired by a letter received from Grace Bedell, a young girl who saw him speak on 15 October 1860 (the story is recounted here in this 1966 NY times article). She proposes the following to the future president (a scan of the original letter can be found here):
I have yet got four brothers and part of them will vote for you any way and if you let your whiskers grow I will try and get the rest of them to vote for you you would look a great deal better for your face is so thin. All the ladies like whiskers and they would tease their husbands to vote for you and then you would be President.
Lincoln's polite response dated 19 October 1860, just prior to the election, is here (source - Library of Congress).
As to the whiskers, having never worn any, do you not think people would call it a silly affectation if I were to begin it now?
I guess to get back to your question, it looks like Lincoln decided to radically change his appearance, probably some time between late October 1860 (before the election) and early January 1861 (post-election but before the inauguration). This is notable too, for the fact that he was the first sitting president to sport facial hair (at least from my informal examination of the official presidential portraits - which you can find here and here). Not sure what the reaction was in the public but I can try to sleuth some more sources.
Hope this is an ok answer.
[*small edits for typos and clarity]