To facilitate the understanding of Invar effects and design of FeNiCo-base Invar alloys characterized by low thermal expansion coefficient (TEC), we investigated the magnetic and thermal expansion behavior of an equiatomic prototype medium entropy alloy FeNiCo and a non-equiatomic (super Invar)Fe₆₃Ni₃₂Co₅ (at. %)reference alloy by means of experiments and ab initio calculations. Both alloys consist of a single face-centered cubic phase with fully recrystallized microstructure in the homogenized state. Large spontaneous volume magnetostriction is observed in both alloys below their respective Curie temperatures. The Invar effect in the non-equiatomic Fe₆₃Ni₃₂Co₅ alloy is of step-type with nearly zero TEC over a wide temperature range (from room temperature to 120 °C)below its Curie temperature. The equiatomic FeNiCo alloy shows a peak-type Invar effect in a very narrow temperature range (from ∼675 °C to ∼730 °C)with relatively low TECs. The equiatomic FeNiCo alloy shows both higher saturation magnetization and Curie temperature than the non-equiatomic Fe₆₃Ni₃₂Co₅ alloy. The relationships among magnetic behavior, spontaneous volume magnetostriction and Invar effects for a wider array of metallic alloys are discussed mainly based on Masumoto's rule combined with Wohlfarth's itinerant electron theory. An Invar alloy search map is constructed based on the present results and available literature data to visualize the relationships among saturation magnetization, Curie temperature and thermal expansion coefficient for a wide range of Invar alloys. Based on this treasure map a design route for further developments of new Invar alloys by tuning their magnetic properties is discussed.