Generally the case.
However, that will be pertinent only for the cases where either second segment WAT (reading your comment precisely) or early second segment obstacle (a bit more generally) is the problem. Both will predispose toward a lower acceleration height to get that last kilo or so.
But you are looking at only two amongst many potentially limiting cases under consideration for any given takeoff .. so, it still depends on this and that.
So far as having a "standard" acceleration height is concerned, it usually comes back to the operator's flight standards views regarding SOP.