I am trying to follow the proof here for Proposition 6 in https://terrytao.wordpress.com/2010/01/03/254a-notes-1-concentration-of-measure/, but I cannot figure out how the computation for the final part should be done:
Each ${X_{i,m}}$ is clearly bounded in magnitude by ${2^m}$; from the subgaussian hypothesis one can also verify that the mean and variance of ${X_{i,m}}$ are at most ${C' \exp( - c' 2^{2m} )}$ for some ${C', c' > 0}$. If ${A}$ is large enough, an application of the Chernoff bound (11) (or more precisely, the refinement in Exercise 3) then gives (after some computation) $$\displaystyle {\bf P}( |S_{n,m}| \geq 2^{-m-1} A n ) \leq C' 2^{-m} \exp( - c' A n )$$ (say) for some ${C', c' > 0}$, and the claim follows.
What I've tried: Each $X_{n,m}$ is bounded by $K=2^m$. We also have $\sigma=$ square root of the variance of $S_{n,m} \leq \sqrt{nC'}\exp(-c'2^{2m}/2)$. Finally define $\lambda := 2^{-m-1} A n /\sigma$. We apply the improved Chernoff's bound as suggested: \begin{align*} {\bf P}( |S_{n,m}| \geq 2^{-m-1} A n ) &= {\bf P}( |S_{n,m}| \geq \lambda\sigma)\\ &\leq C \max( \exp( - c \lambda^2 ), {(\lambda K/\sigma)^{-c \lambda \sigma / K}} )\\ &\leq C \max\left(\exp(-c\cdot 2^{-2m-2}A^2n\exp(c'2^{2m})/C''),(A\exp(c'2^{2m})/2C')^{-2^{-2m-1}cAn} \right)\\ &\leq C'' \exp(-c''An)(A/2C')^{-2^{-2m-1}cAn}. \end{align*} $(A/2C')^{-2^{-2m-1}cAn}$ tends to $1$ as $m$ is large. I cannot get the $2^{-m}$ factor to show up in the final step.
My guess is that $P(|S_{n,m}| \ge 2^{-m-1}An)$ is a typo. The first piece of evidence to support my guess is that right above in the proof, one encounters $P(|S_{n,m}| \ge \frac{A}{100(m+1)^2}n)$. The second piece of evidence is that, as you correctly showed, the proof doesn't work with $P(|S_{n,m}| \ge 2^{-m-1}An)$ instead.
Let $\lambda = \frac{1}{100(m+1)^2}\frac{An}{\sigma}$. Then $P(|S_{n,m}| \ge \frac{An}{100(m+1)^2}) = P(|S_{n,m}| \ge \lambda \sigma) \le C\max(\exp(-c\lambda^2),(\lambda K/\sigma)^{-c\lambda \sigma K})$. Note $\exp(-c\lambda^2) = \exp\left(-c\frac{1}{m^4}A^2n\exp(c'2^{2m})\right)$ is obviously at most $C'2^{-m}\exp(-c'An)$. And the term that was problematic for you, i.e. $(\lambda K/\sigma)^{-c\lambda \sigma/K}$, is now about $(\frac{2^m}{m^2}A\exp(c'2^{2m}))^{-c\frac{An}{m^2}2^{-m}} \approx \exp(-c\frac{An}{m^2}c'2^m)$, which is also clearly small enough.