I realise that Gram points can approximate the imaginary part on the $x$th zeta zero $(\gamma_x)$ accurately, and indeed, Guilherme França, André LeClair give another formula, namely $$\dfrac{2\pi(x-\frac{11}{8})}{W(e^{-1}(x-\frac{11}{8}))}$$ in this paper, where $W(x) = $ Lambert W function.
$$$$ I noticed that the parametric plot of $\{\log\Gamma(x),2\pi x\}$ was also very similar to $\{x,\gamma_x\}$.
Approximating the inverse gamma funcion ($\Gamma_1^{-1}$) as per here $\gamma_x$ can then be approximated with
$2\ \pi\ \Gamma_1^{-1}(\operatorname{exp}(x-1))-\pi$, and gives a surprisingly good fit, even for very high $\gamma_x:$
ie, $\gamma_x\sim$ $$ 2\pi\log\left(\dfrac{\operatorname{exp}(x-1)+\frac{\sqrt{2\pi}}{e}-\Gamma(k)}{\sqrt{2\pi}}\right)\bigg/ W\left(\frac{1}{e}\log\left(\dfrac{\operatorname{exp}(x-1)+\frac{\sqrt{2\pi}}{e}-\Gamma(k)}{\sqrt{2\pi}}\right)\right) $$
where $k\approx 1.461632\dots$
Alternatively, it could be stated $\gamma_{\lfloor\log\Gamma(x)\rfloor}\sim 2\pi x$.
Is there anything different being conjectured here, or is it basically a restatement of the above?
k=x/.FindRoot[PolyGamma[x]==0,{x,1},WorkingPrecision->20];
g[x_] := 2 Pi Log[(Exp[x - 1]+Sqrt[2Pi]/E-Gamma[ k])/Sqrt[2Pi]] /LambertW[(1/E)Log[(Exp[x - 1]+Sqrt[2Pi]/E-Gamma[k])/Sqrt[2Pi]] ]
Update
Interestingly, $\dfrac{\gamma_{\lfloor\log\Gamma(x)\rfloor}}{2\pi x}$ appears to converge to $1$ at a similar rate to $\left(1+\frac{1}{\text{const.}x}\right)^{\text{const.}x}\rightarrow e.$
I include a plot of $\dfrac{\gamma_{\lfloor\log\Gamma(x)\rfloor}}{2\pi xe^{-1}}\bigg/\left(1+\frac{1}{0.83 x}\right)^{0.83 x}$ below:
