Writing an acceleration formula that is dependent on displacement

Question

So, for an assessment task, I wanted to model the height of a rocket above Earth. I created a piecewise acceleration formula, the first part of which is when the rocket accelerates, the second part of which is just gravity. This can then be integrated twice to find a displacement formula. However, gravity varies with distance from the centre of the Earth, and with my chosen values, this ended up making the rocket peak twice as high (height above sea level) than what was predicted. The problem is that if I change gravitational acceleration from constant to using the gravitational field formula, gravity is proportional to $\frac{1}{r^2}$. $r$ can only be found with the twice integrated acceleration formula. So, from what I understand, I would have to put in a dummy value for displacement into the acceleration formula, integrate for displacement, then substitute that back into the acceleration formula, then repeat the process to achieve higher fidelity. Is there a better way to do this? Otherwise, I can just have a nice evaluation at the end of my assessment task. I think the maths I've used is already good enough to express my skill for good marks at the level of my course.