Yes, at least if your chosen element is the lexicographically first reduced expression for your permutation.  This is how people usually prove that any two reduced expressions for a given permutation are connected by a series of long and short braid moves -- by reducing both to the lexicographically first reduced expression for that permutation.  This is done by greedily applying $s_j s_i \rightarrow s_i s_j $ for $|j-i|>1$ and $s_{i+1}s_is_{i+1} \rightarrow s_is_{i+1}s_i $ moves until no further such moves are possible.

You can find such an algorithm for the symmetric group e.g. in:

Adriano Garsia, The saga of reduced factorizations of elements of the symmetric group, Publications du Laboratoire de Combinatoire et d' Informatique 29 (2002)

While oddly enough, I don't see this book listed on MathSciNet, I just found a copy available for free by googling: Adriano Garsia saga

This is also discussed (for more general Coxeter groups) in the book:

Anders Björner and Francesco Brenti, Combinatorics of Coxeter groups, Graduate Texts in Mathematics, 231, Springer, New York, 2005, xiv + 363 pp.

At least the connectedness result is there, and I believe it is again by this type of algorithm -- I don't have the book with me right now to check for sure.  

Another potentially relevant reference is:

Paul Edelman, Lexicographically first reduced words, Discrete Math, 147 (1995), no. 1-3, 95--106.