My understanding of the tradition of the Vilna Gaon (whose tradition is followed in Israel) with respect to shir shel yom is that every day has exactly one shir shel yom. For example, during Hanukkah, the shir is Psalm 30 and on Rosh Hodesh it's Psalm 104 instead of the regular shir shel yom.
So my question is on Rosh Hodesh Tevet (which is also Hanukkah), what is the shir according to this tradition?