Recall that a tree on $\omega_1$ is called Suslin if every chain and antichain are countable. If every level is countable and there are no cofinal branches, then it is called Aronszajn (in particular Suslin is Aronszajn). Moreover, an Aronszajn tree is called special if it is the countable union of antichains.
So special Aronszajn trees and Suslin trees represent some sort of endpoints on the spectrum of Aronszajn trees: either every antichain is countable, or the whole tree is a countable union of antichains.
Given an Aronszajn tree, we can always specialize it using a ccc forcing; and given a Suslin tree it is a ccc forcing by itself, and forcing with it will add a cofinal branch (which is an uncountable chain).
Therefore, given a Suslin tree, we have a choice of how we want to violate its Suslinity: either violate its Aronszajn-ness, or specialize it.
Can we make a Suslin tree into a non-special Aronszajn tree, while destroying its Suslinity?
If the answer is positive, what is the "best" way of doing so? (Either consistently or provably, from the existence of a Suslin tree.)
Can we do it with a ccc forcing, or with a $\sigma$-closed forcing, or with just a proper forcing? Do the properties of this forcing somehow depend on the tree (e.g. if the tree is rigid, or homogeneous, etc.)?
(As a minor bonus question, is it consistent that all Aronszajn trees are Suslin or special, but Suslin trees exist?)
Just a remark on triviality, when I say "tree" I always mean that any point has at least two successors, and that it has successors on every level of the tree. I am also going to assume the tree is Hausdorff, in the sense that at limit levels the points are always determined by their predecessors.