How could I improve this approximation?

Bhaskara's sine approximation looks a bit like a Padé approximant and is eerily accurate.

Another option---since you need to do this trillions of times---is to save the solutions for many values of $a$, and then look up/interpolate solutions as you need them.

We have to solve the equation ${\rm sinc}(x)=a$ for $x\in[0,\pi]$ in terms of the parameter $a\in[0,1]$. As Kirill has remarked, for $a\to1\!-\ $ we obtain $x\doteq\sqrt{6(1-a)}$. Therefore, if we want a single simple expression giving accurate values over the whole $a$-interval $[0,1]$ we have to take a square root at the end.

For this reason we put $x:=\sqrt{u}$ and solve the equation $$\bigl(s(u):=\bigr)\quad {\rm sinc}\bigl(\sqrt{u}\bigr)=a\tag{1}$$ for $u$ in terms of $a$. Let $$a\mapsto u:=f(a)\qquad(0\leq a\leq1)$$ be the solution of $(1)$, i.e. the inverse function of $s$. From known values of $s$ and $s'$ we can, e.g., deduce the values $$f(0)=\pi^2,\quad f\left({2\over\pi}\right)={\pi^2\over4},\quad f\left({\sqrt{27}\over 4\pi}\right)={4\pi^2\over9},\quad f(1)=0\ ,\tag{2}$$ and $$f'(0)=-2\pi^2\tag{3}$$ (one could take more such values into account and determine more coefficients $c_k$, $d_k$ in the following).

We now make the "Ansatz" $$\tilde f(a):={p(a)\over q(a)},\qquad p(a):=\pi^2+c_1 a+ c_2 a^2,\quad q(a):=1+d_1a +d_2 a^2\ ,$$ and determine the coefficients $c_1$, $c_2$, $d_1$, $d_2$ such that $(2)$ and $(3)$ are satisfied. This leads to the approximation $$x=g(a):=\sqrt{\tilde f(a)}=\sqrt{{\pi^2+5.95839 a - 15.828 a^2\over 1 + 2.60371 a + 0.690687 a^2}}\ .$$ The following figure shows a plot of $a\mapsto \sin\bigl(g(a)\bigr)-a\> g(a)$:

Hi intersting question usimg hermite polynomial $$\text{Solve}\left[\sum _{n=0}^3 \frac{\left(i (-1)^{2 n} i^n 2^{-n-1} \left((-1)^n-1\right)\right) H_n(x)}{\sqrt[4]{e} n!}=a x,x\right]$$ you could get $$\left\{\{x\to 0\},\left\{x\to -\sqrt{\frac{3}{2}} \sqrt{5-4 \sqrt[4]{e} a}\right\},\left\{x\to \sqrt{\frac{3}{2}} \sqrt{5-4 \sqrt[4]{e} a}\right\}\right\}$$ give arounf three digits of precistion