It is worth noting that the text of Hamlet given on the website linked to above is defective. Admittedly the play is notorious for its textual problems, but it is careless, to say the least, for the site owners not to indicate the sources of their text.
This would not matter so much perhaps, except that in the very song under discussion, the second to last line is wrongly given as:

"Which bewept to the grave did go"

This should read the opposite:

"Which bewept to the grave did not go"

This reading appeared in all the original texts (Quarto 1 &2, and the First Folio),but was "corrected" by Alexander Pope as being unmetrical.
Why Shakespeare should have inserted this extra syllable into the line is a matter for interpretation; perhaps it is a pointed comment by Ophelia that her father was buried in secrecy "hugger-mugger", without anyone to mourn him.

Some of the other songs in this scene are intriguing.Tomorrow is Saint Valentine's day is a typical night-visiting song with the usual roles reversed in that is is the female who appears at the window pleading to come in.The fragment:

They bore him bare-faced on the bier
Hey non nonny, nonny hey nonny,
And in his grave rained many a tear
[A-down, a-down, a-down-a]
has the ring of an old ballad, especially in the alliteration of the first line;bore/bare/bier. The Bonny sweet Robin comes from a common 16th century ballad, while the dirge And will a not come again? is another otherwise unknown song.
Anyway, good luck with your searches Ian.