Certainly, the plot you suggest can work; but I ask you whether you have given thought to the relation between the speed of light and the size of our Solar System.
Light takes eight minutes to travel from the Sun to the Earth, a distance of about ninety million miles. Mars is something like another forty million miles farther out from the Sun than we are. If Mars were at the opposite side of the Sun from Earth at the time a ship took off at lightspeed times three, allowing for a wide arc to clear the Sun safely, the ship would have a little over two hundred fifty million miles to travel. Assuming that the ship was able to make its top speed from the very moment of activating its drive, it would cover the whole distance in half an hour or so.
But ARE you saying that the ship can be instantly at lightspeed times three? It looks as if that's what you mean. Now, do you realize that there are asteroids in the Solar System? Do you really want to be flying faster than light in an environment so dense with obstacles? That would be like driving a car through a forest full of thick trees at 100 miles per hour with your eyes closed. Even Han Solo was flying slower than light when he navigated an asteroid field.
On the other hand, maybe you intend to be more realistic, and say that your ship needs time to accelerate to faster-than-light speed. But if there is an acceleration process, then in order to reach that maximum speed without destroying everyone inside the ship, it would take TIME building momentum. So much time, that there would be no point in going up to lightspeed times three. By the time the ship even got up to, say, one-quarter lightspeed, it would already be going fast enough to make the whole journey with marvellous quickness.
If you are going to write science fiction, you need to do your homework. I am not saying this to make you give up, but to help you do better.