This paper presents the design and operations of a compact cryogenic scanning tunneling microscope system combined with a dilution refrigerator. We obtained a minimum temperature of 260 mK at the sample position. Taking advantage of low-temperature measurement for the spectroscopy, inelastic electron tunneling spectroscopy (IETS) at 4.4 K was demonstrated for the octanethiol molecules in a self-assembled monolayer. The spectrum showed many vibrational features as in the case of a high-resolution electron energy loss spectroscopy. We discuss the resolution of the IET signals focusing on its changes with the modulation voltage and the sample temperature. IETS at 260 mK is also presented with similar quality as in the case of 4.4 K, indicating that the vibration-dumping methods for the operation of the dilution refrigerator described in this paper are adequate.