The measured soft X-ray, self-seeding spectrum at the LCLS free-electron laser (FEL) often has a pedestal-like distribution around the seeded wavelength, which limits the spectral purity and can negatively affect some user applications not employing a post-undulator monochromator. In this paper, we study the origins of such pedestals, focusing on longitudinal phase space modulations produced by the microbunching instability upstream of the FEL undulator. We show from theory and numerical simulation that both energy and density modulations can induce sidebands in a high-gain, seeded FEL whose fractional strength typically grows as the square of the undulator length. The results place a tight constraint on the longitudinal phase space uniformity of the electron beam for a seeded FEL, possibly requiring the amplitude of long-wavelength modulations to be much smaller than the typical incoherent energy spread if the output sideband power is to remain only a couple percent or less of the amplified seed power. Show Partial Abstract