Actually if you look close your EI3 is in reality a light stamp EIG.
The EIG logo was an importer in Miami, Fl owned by Saul Eig. in the mid 60's early 70's
I'd agree it might have been made by Pietta , ASM , Grassi,Doninelli, and Gazzola or one of a few other cottage gun shops in Brescia.
I see the Gardone Proof marks and if you look you should find a Roman Numeral date code example: XX or XXI XXII etc.
The bent coat hanger symbol is an early B inside G for Brescia, Gardone ( Brescian: Gardù de Altrompia ) , home to many Arms makers , including Beretta , Uberti, Pietta , Benelli , Fabarms and at the time your gun was made dozens of Cabin shops and many fabricators of arms artisans... ( if you find XXVI = 1970 good chance its a Grassi,Doninelli, and Gazzola manufacture.)