If you have private coverage at home, as a non-EU citizen, the requirements are different. In order for your private health insurance in Italy to be valid and to obtain a permit to stay, some requirements include but are not limited to: legitimacy in the Schengen area, exact coverage period and expenses and a ceiling of 30,000 euros.
If you choose to continue with private health insurance in Italy, it is recommended that you invest in the INA-Assitalia Insurance, which accounts for urgent medical assistance in the Emergency Room of any hospital. This costs around 49 euros for six months and 98 euros for one year.
