No. The only advantage of working for a big company is that it is very likely
to find people who worked with you for reference. And big companies stay
around longer, so hiring company can always go back to check.
Nobody looks people down from small companies. Nobody cares what tags your
used before. Your work experience and skills worth millions times more. One
difference though is that management titles do not mean the same. One maybe
the CEO of a startup but only can be a director in a big com