Is this a good time to go for higher studies (in management or similar)?

I'm a final year student at a tier-1 engineering college. I am looking to break into product management and have been looking for internships and full time opportunities for the last 5 months. I've had no luck so far, and my campus placements are not up to the mark this year so no luck there as well. I've done all the essentials for preparing for this role, like preparing for interviews, participating in product teardowns, working on a portfolio etc. I initially planned to work for a few years and do an MBA in the US but I'm seriously contemplating going for a master's degree now (maybe MiM in EU or some MS in software applications in the US). Looking for advice on what should be my next steps. PS: I'm not looking to do an MBA in India. If I go for a master's degree, I want to do it abroad.