It means " I can't be more happier" in this sentence. Sometimes it means "I don't have enough time to do something" There is a romance novel call " 来不及说我爱你“ means " I don't have enough time to say I love you" Sometimes it means too late to chatch up/ do something like "你再不快一点,上班就来不及了” means If you don't hurry up, you can't go to work on time."

I'm more than happy to have such beautiful hair. 來不及 in this sentence means "you're too happy to think about anything else." Literally, you only have time to be happy and have no time for other things. More examples: 失去了心愛的人，他傷心都來不及了（怎麼還會想再去約會呢） 得到了一千萬，他開心都來不及了（怎麼還會想再繼續工作呢） Hope it makes sense!