Solving numerically a non-Born-Oppenheimer time-dependent Schrödinger equation to study the dissociative-ionization of H₂ subjected to strong field six-cycle laser pulses (I = 4 × 10¹⁴ W/cm², λ = 800 nm) leads to newly ultrafast images of electron dynamics in H₂⁺. The electron distribution in H₂⁺ oscillates symmetrically with laser cycle with θ + π periodicity and gets trapped between two protons for about 8 fs by a Coulomb potential well. Nonetheless, this electron symmetrical distribution breaks up for the H₂⁺ internuclear separation larger than 9 a.u. in the field-free region at a time duration of 24 fs as a result of the distortion of Coulomb potential where the ejected electron preferentially localizes in one of the double-well potential separated by the inner Coulomb potential barrier. Moreover, controlling laser carrier-envelope phase θ enables one to generate the highest total asymmetry A_e^tot of 0.75 and -0.75 at 10^○ and 190^○, respectively, associated with the electron preferential directionality being ionized to the left or the right paths along the H₂⁺ molecular axis. Thus the laser-controlled electron slightly reorganizes its position accordingly to track the shift in the position of the protons despite much heavier the proton’s mass.