The best time to build a resume is when you are still in school. Why do you want to build a great resume? Because you want to have an easier time getting an offer for a great job, ideally before you graduate. In my experience, working hard at looking for a job after graduation while living hand to mouth - that's a bummer of a life experience.
When I was at uni, I had constant money problems due to my family's financial situation, and I had to do my part to support them. In my context, I'd take a paid part-time job at a small company rather than unpaid internship at a big name organization.
I understand that Google, Amazon and Facebook internships are paid gigs and I go for paid gigs - my attitude back then would have been, even if the internship were a bust, I'd still get paid. You'd get exposure to people and technologies that is simply not available in small companies. And of course, opportunity comes with exposure. It's much easier to argue that you can do it because you've done it than to argue that you can do it even though you've never done it. "Yeah, I was scared of it but after that internship where I was exposed to it and I was neck deep into it, I learned that fear is just a state of mind. Now, I just don't give a damn. Give me an elephant of a problem and I'll figure out how to eat the damn thing one bite at a time".
You can get high quality if narrow exposure at a small firm depending on the firm's needs and your sense of initiative. I started out as a part-time engineer and computer programmer at a seven-person startup and I played a critical role in this start-up becoming the largest environmental firm in the New York Metro area.
At the end of the day, it's you - what you learn, what you do, how you apply what you learn to do what you do. It's all about you, your level of initiative in making your job into something more than it is, your guts in being willing to make mistakes that you can learn from and you being able to convey that the reason you are confident in the face of challenges is that you've taken so many of them. And won despite moments when you felt like shitting your pants and even then, shitting your pants wouldn't have stopped you.
