  We reanalyze the factorization theorems for Drell-Yan process and for deep
inelastic scattering near threshold, as constructed in the framework of the
soft-collinear effective theory (SCET), from a new, consistent perspective. In
order to formulate the factorization near threshold in SCET, we should include
an additional degree of freedom with small energy, collinear to the beam
direction. The corresponding collinear-soft mode is included to describe the
parton distribution function (PDF) near threshold. The soft function is
modified by subtracting the contribution of the collinear-soft modes in order
to avoid double counting on the overlap region. As a result, the proper soft
function becomes infrared finite, and all the factorized parts are free of
rapidity divergence. Furthermore, the separation of the relevant scales in each
factorized part becomes manifest. We apply the same idea to the dihadron
production in $e^+ e^-$ annihilation near threshold, and show that the
resultant soft function is also free of infrared and rapidity divergences.
