After years of research (;-)), I might have found the answer in Oxford's Advanced Learner's Dictionary (1974):
1. Too (also; as well, in addition)
Usually with end position but placed immediately after the word it modifies if there is a risk of ambiguity.
Examples:
I, too, have been to Paris. (I, as well as he, you, etc.)
I've been to Paris, too. (to Paris as well as to Rome, Milan, etc.)
She plays the piano, and sings, too. (plays the piano and also sings)
Sally, too (=Sally, as well as Mary, etc.) plays the piano.
2. Too (moreover; nevertheless)
There was frost last night, and in May too!
It seems that I have put the word in the wrong place all the time, but the comma has always been there!