Our final equations for the position and the velocity become then $$x_{i+1} = x_i+hv_i+\frac{h^2}{2}v^{(1)}_{i}+O(h^3),$$ and $$v_{i+1} = v_i+\frac{h}{2}\left( v^{(1)}_{i+1}+v^{(1)}_{i}\right)+O(h^3).$$ Note well that the term $v^{(1)}_{i+1}$ depends on the position at $x_{i+1}$. This means that you need to calculate the position at the updated time $t_{i+1}$ before the computing the next velocity. Note also that the derivative of the velocity at the time $t_i$ used in the updating of the position can be reused in the calculation of the velocity update as well.