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At the end of Death Note, no one knows that Light still has Death Note paper on him. So, he could have waited to be in the car or in prison to use the Death Note. That would have been the smartest thing to do in the situation he was in.
So why didn't Light Yagami go to prison and then write the names of the NPA and the SPK?
There are several reason why the story didn't follow that path.
Firstly, I believe this is in part due to the fact that Light did not want to lose the "game". Throughout the entire first half of the series, there was a lot of emphasis put onto the fact that L needed to be 100% certain that Light Yagami was Kira before he'd authorize the arrest of Light.
Even near the end of the first half, when L was almost 100% certain, he still wouldn't act on this intuition.
Secondly, despite the fact that he was detained by L for an extended amount of time, Light knew that he'd eventually be released within a reasonable amount of time, but if he were to be arrested, he'd be locked up for years & stripped of all of his possessions, so even if he hid it inside a watch like he had previously, the officers would have confiscated it, leaving Light without a scrap of paper from the death note. Then, there is the fact that once he got incarcerated, unless he could manage to set up some brilliant break out ahead of time, he'd have no way to escape (if they didn't just outright execute him for his crimes). To top all of that off, if he were to escape, he'd be a known escaped convict & hunted for the rest of his life.
Thirdly, in the last episode of Death Note, Light simply still didn't know the name of Near. Only Teru Mikami ever saw Near's real name, and despite him writing it down, Light never got the chance to see the notebook that Teru wrote the name on.
Sorry that this is getting quite long-winded, but I have one last fact to point out that I think will be the most important one. This is brought up at the very beginning of Death Note, as well as in the very last episode. In the very beginning, Ryuk states that when Light dies, he'd be the one to write Light's name down. So, since we know this and since we know that Ryuk did, in fact, write Light's name down on his death note in the last episode, we can conclude that Light was meant to die there, regardless of what happened. Since the fact that lifespans can be seen about someone's head to know when they were going to die, this institutes that a person's lifespan is set unless a shinigami steps in, like in the case of Misa Amane. Therefore, the fact that Ryuk wrote Light's name down in his book at the end meant that no matter what happened, Light was going to die in that moment.
