Hats off, Bogie. The traditional Italian hat maker Borsalino, whose fedora has defined the rough-and-tumble images of Robert Redford, Frank Sinatra and Humphrey Bogart, has a plan to appeal more to women and millennials by pushing into high-end fashion, streetwear and even sportswear.

The recent troubles of the company, which has been rescued financially by a private equity fund, underline the difficulties facing Italian companies — not only fashion brands — seeking scale to stay competitive and grow in the age of globalization. Entering new markets, targeting new demographics and expanding retail presence all require cash. And even the best-laid business plan can be complicated by a company’s bad baggage, and Borsalino came with plenty of that.