Compost amendment is a widespread agronomic practice, but little information is available about the short- and medium-term effects on clay soils. In this investigation, we selected three soil compost rates (treatments, T), i.e., fertilizer (T2 = 1.5 kg m−2), amending (T3 = 15 kg m−2), and organic (T4 = 75 kg m−2), that were compared to a control (T1). Our research accounts for the effects reached on representative large boxes (about 0.75 m3), without crops and for about eighteen months. An overall assessment of the physical and hydraulic properties of the soil was made, including infiltration rate or saturated hydraulic conductivity (Ks), water content and water retention (θ) of the soil, bulk density and structure, and several physical quality indicators obtained from the water retention curve, accounting for the optimal balance between water/air into the soil, pore volume distribution function, and soil features in the inflection point of the soil water retention curve. Additionally, the temporal changes of Ks were evaluated. The main results showed that (i) after eighteen months, and regardless of T, θ significantly improved by a factor of 1.2–1.3, but these improvements (up to 1%) were detected only close to water saturation (i.e., until 6 cm of soil pressure head) when little (T2) or no compost (T1) addition was considered, while a larger range was detected (until 60 cm) when higher rates (T3–T4) were used; (ii) Ks determination allowed to establish that compost effects vanished after about eleven months, but it was not possible to verify that composting increased the permeability of the investigated clay soil within that time frame; (iii) some significant correlations between Ks and some soil physical indicators estimated from both the inflection point of the water retention curve or bulk density suggested possible improvements in soil permeability. Because some factors (water content of the soil above all) could have affected the comparison of Ks measurements, further research on this topic is needed.