Thank you for posting the second image.
At the beginning of this we have the word assuredlie, with a long-s in the second position.  
The shape and plainness of the bottom part of the long-s can be observed.
The word above Elizabeth's surname is saved, with a long-s in the first position.
I believe a similar plain shape can just be discerned above the surname.  It is fainter and less clear than an additional squiggle which partly overlays it.
It's possible but not certain that the additional squiggle includes a letter a.
The squiggle is probably a contraction mark, and what has been contracted is probably the letters ia.
That makes the surname will(ia)mson.
However it's not an open-and-shut case and others may disagree.