Zero Free Region of Riemann Zeta Function for the line $\Re(s)>1-\epsilon$

Question

I am familiar with the classical zero-free region of $\zeta(s)$. That is if $\rho=\sigma+it$ is a non-trivial zero of the Riemann zeta function the classical zero-free region says that there are no zeros in the region $$ \sigma\geq 1-\frac{c}{\log t} $$ I know that this bound implies that there are no zeros on the line $\sigma=1$. However, I am a bit unsettled with this since for any $\epsilon>0$, there is possibly for $t$ large enough such that a zero $\sigma+it$ may occur with $1-\epsilon<\sigma$. Hence with such a region as long as the imaginary part is large enough it is possible to find zeros that are arbitrarily close to the line $\sigma=1$. I am curious if there are any results that actually move the line of the zero-free region away from $\sigma=1$. That is has it been shown that there is some $\epsilon>0$ such that there are no zeros of $\zeta(s)$ with $\sigma>1-\epsilon$. If such a result exists, I'd love a reference since I would expect such a result to be highly non-trivial.