This paper is a continuation of a previous work of the author [M. Wollenberg, Lett. Math. Phys. 31, 195-203 (1994; Zbl 0812.46073)], considering the question how the conformal structure of spacetime can be reconstructed from nets of local observable algebras. The basic idea of the author is to use algebras \({\mathcal A}(\gamma)\) associated to smooth curves \(\gamma\) to distinguish spacelike and timelike curves. In the above mentioned previous paper, this is done by the assumption that \({\mathcal A}(\gamma)\) is abelian if \(\gamma\) is spacelike and not abelian if \(\gamma\) is timelike. However, in the case of a free Klein-Gordon field on a globally hyperbolic spacetime this can be checked only in some special representations. Therefore the properties of the \({\mathcal A}(\gamma)\) (especially for the Klein-Gordon field) are reconsidered in the present work, and several new possibilities to determine the causal character of \(\gamma\) are discussed. The paper concludes with a new scheme which is applicable to the free Klein-Gordon field in any faithful representation.