What else does Constant's will tell us, apart from the name of his ship, and the name of his captain, and the fact he was getting ready to make another voyage to the Far East?

He had a wife.  Her name was Mary, and she lived in a part of London called Poplar.

(You won't be surprised to learn that that Poplar, which was really only developing in Constant's time, is on the Thames. The Thames was more than a source of expensive views to John Constant's London.  It was as important as Heathrow and Gatwick put together, it was a superhighway, a great sea highway through which the riches of empire flooded into the city.  London was first and foremost a port.   Constant was a sailor, and he, his wife and possibly his captain, Henry Smedley, would want to live close to the river.)

All Saints Church, Poplar

So here is the first mention of Mary Constant in her husband's will:

                              "..say, all my Wages Sum and Sums of Money  Lands

                            Tenements Goods Chattles and Estate whatsoever as shall

                            be any ways due owing or belonging unto me at

                            the time of my decease I do give devise and bequeath

                            the same unto my Wife Mary Constant of Poplar in 

                            the County of Middlesex And I do hereby nominate and..." 

..And  he goes on to name his wife Mary as his sole executrix.  John is facing the fact he may not come back from his voyage, and so he is careful to leave everything to the one person he wants to provide for.  There are no bequests to his parents or to any siblings, as often there were.  But then, how could there be?  John had been snatched away from those a long time since.