Yes.  Choose a reduced expression $red(w)$ for $w$.  Now delete from $red(w)$ the rightmost letter $s_i$ that appears more than once in $red(w)$.  This will yield a reduced expression for an element $v\in W$ with the same support as $ w$ and length one less than $w$, with $v$ less than $w$ in Bruhat order.  Repeat until you have decreased the length to $r+1$.