I think that $A=2d$ will work, basically by applying Morse theory to the distance function from $x$. Morse theory for distance functions was originally considered by Gromov (and then Cheeger). For the case of surfaces, see a paper of Gershkovich (also see Gershkovich-Rubinstein for a Morse theory of distance functions).

Imagine taking metric balls $B_r(x)$ of radius $r, 0\leq r\leq d$ about $x$, so that $B_d(x)=S$. These will be the image of a hyperbolic disk $D_r$ at $\tilde{x}$ of radius $r$ in the universal cover $\mathbb{H}^2 \overset{\rho}{\to} S$. For $r$ sufficiently close to 0 ($\leq$ the injectivity radius at $x$), $B_r(x)$ will be an embedded image of the disk $\rho(D_r)$. When $r=injrad_x(S)$, there will be some tangencies of pairs of points in $\rho(\partial D_r(x))$. Taking geodesics in $D_r$ from $\tilde{x}$ to these pairs of points in $\partial D_r(x)$, and projecting by $\rho $ to $S$, we see a collection of loops of length $2r$ based at $x$. Now, continue to let $r$ increase. Each time we see a pair of tangencies between pairs of points in $\partial B_r(x)$, create a "dual" loop. Once one reaches $r=d$, we will obtain a collection of loops cutting $S$ up into disks (one may argue this via Morse theory: self-tangencies are the only way to create 1-handles). Then taking a subset of $2g$ of these loops, one obtains a standard collection, each with length $\leq 2d$.