How should one expand the information set in VAR models?
My recommendation is that you read through this paper from Bernanke, Boivin, and Eliasz. This paper introduced something called factor-augmented VAR. The gist is that you can use principle component analysis to do exactly what you're after. This method allows you to include hundreds of time series in the conditioning information. And FYI, this does allow one to solve the price puzzle with solid economic rationale instead of the ad hoc inclusion of a single time series.