As Joel points out, this is not possible. Even in the case of a discrete series rep'n, one can't control the value of the central character on the non-compact part of $F_v^*$. What one *can* do (in some generality; for precise references recent work of Sug Woo Shin might be relevant) is show that the set of tempered representations which *are* local components of automorphic forms are "large" in the space of all tempered rep'ns, where "large" can be taken to mean something like "equidistributed according to Plancherel measure", or also "Zariski dense in the Spec of the Bernstein centre". (There may well be caveats to both these statements, though, so I would look at Shin's work and whatever references are contained therein.)