Do you know where the woman was living when she divorced. In the 60s as I remember the local papers contained listed from time to time of decree nisi in the local court including the grounds. Means a long trawl possibly but could be successful,depends what you want/expect to find. The short length of the marriage suggests adultery or failure to consummate because in the 60s divorce was a much more lengthy process otherwise

With regard to the re-marriage, it may not be much help but there is another possibility, She used the name of the man she married because she was already known by that name. In the 60s you could certainly say you wished to be known by any name you chose as long as there was no fraudulent intent. Again the local papers sometimes carried a formal public notice to that effect presumably paid for by the person concerned. Incidentally you also saw notices that a man would not be responsible for a wife's debts presumably because she'd run off with another man in most cases. Identity theft was not a problem and you didn't have to give innumerable details to get anything done. I would think you just gave notice in the registry office, which is where most divorced people remarried, with whatever details you wanted probably producing the decree absolute and signing that the information you gave was correct and that was sufficient. So the certificate might say X otherwise known as Newton formerly Boot but only X would appear in the index.