It's well known that, if GCH holds, then every cardinal can be well-ordered. However, I'm sure we don't need full power of GCH to prove it for specific cardinal, e.g. continuum. I have been wondering, what is the minimum amount cardinal numbers $A$ for which non-existence of cardinal $B$ with $A<B<2^A$ guarantees that $\frak{c}$ can be well ordered (or, simplier, GCH for which cardinals already implies existence of well-ordering of $\frak{c}$)? By looking through the proof I think it's enough for it to hold for $A\in\{\mathbb{R},2^\mathbb{R},2^{2^\mathbb{R}}\}$.

Thanks in advance for help!