If your push to connect fittings are the type that are uncrimped, you can purchase a push to connect fitting that has a tee off adapter with a ⅜” connection.
If your push to connect fitting is crimped, you can cut the water line and insert a three-way adapter fitting - where two ends are push to connect and the third that tee’s off of the adapter has a ⅜” connection.
If you have questions please take a picture of the connections and give us a call to discuss your installation.