"Have to" is a pretty difficult way to phrase the question, because it implies you can go back in time and re-run things with different choices made and predict the outcomes. It is easier to ask: why did we drop two?

The answer is: because we had two. There was no high strategy to it. It is not clear that Truman was even aware that two bombs would be ready to drop within days of each other. He definitely knew one would be dropped, and all planning discussions centered around the singular use of the bomb. The strike order specified with considerable detail how and when the first bomb would be used, but then had an open-ended clause that said the Manhattan Project personnel could drop more as they had them available. Which it turns out they did. The decision to drop the second bomb, and the timing of it, was essentially left to local military forces on Tinian, because it was seen as an operational decision (as opposed to a strategic one). They dropped the second bomb when they did because they had another bomb ready to go, they foresaw that weather would be bad for visual bombing in the next few days, and then rushed it into use so they wouldn't have to possibly wait. The people who did this did not think it was a big moral or strategic decision.

OK, so what was the effect of the second bomb on the Japanese high command? This is the harder thing to measure, because if there was an effect, it was subtle. After Hiroshima (August 6), the Japanese high command had sent a survey team to the city to confirm it was an atomic bomb (as the American propaganda released 16 hours after the bombing had claimed). This team made its report back to Tokyo late in the evening of August 8th (Tokyo time), confirming it was an atomic bomb. A plan was made for the high command to meet the next day to discuss this. Overnight, the Soviet Union declared war on Japan and invaded Manchuria. So the meeting the next morning (August 9th) was a pretty heavy one — the US had the atomic bomb, the Soviets were invading, things were bad. The Japanese high command, going into the meeting, seemed primed to essentially accept unconditional surrender except for the state of the Emperor (they wanted the imperial house preserved). During this meeting they got news of the bombing of Nagasaki, which had just happened (the mission was already in preparation by the time they had confirmed what had happened at Hiroshima). It does not seem to have overtly affected their thinking during the meeting.

The Japanese high command left that meeting with the understanding that they were offering up a surrender offer — not totally unconditional (the Emperor being the one condition). The US rejected that offer, and began conventional bombing of Japanese cities again. Internally, the Japanese had an attempted military coup, that was put down. Finally, on August 14th, the Japanese agreed to full, unconditional surrender.

I give you all of that context just to show you how many things were going on here. Where does the "signal" of Nagasaki surface above the "noise" of everything else (Hiroshima, Soviet invasion, more conventional bombing, attempted coup, etc.)? It isn't a strong signal (whereas the Soviet invasion was; they clearly were affected by that), but that doesn't mean it didn't contribute to a lot of feelings going on at the time. But it's hard to say, and you can see why some scholars have long suggested that Nagasaki "wasn't necessary" in the sense that you can imagine the exact same result happening without it.

After the fact, people have spent a lot of time trying to justify the Nagasaki bombing (one can ponder why), and the after the fact justification is, "we had to prove to them we had more than one atomic bomb." Which, for one, doesn't seem to be why anyone actually wanted to drop the bomb at the time. But more importantly, they did not think that one or two atomic bombs would probably end the war anyway. General Groves, the person who was perhaps most responsible for both of the bombings, thought it might take 4 or 5 attacks before the Japanese surrendered. So the entire question is one that only comes up when you know that the Japanese were going to surrender soon, which is only knowable after the fact.

What I think one can say with some conviction is that: if the goal was to reduce Japanese civilian casualties (which it wasn't) and the second atomic bomb was only dropped because the Japanese appeared to ignore the first one (which it wasn't, which they didn't), then it was dropped too soon after the announcement of the first one to actually feed into the decision-making of the Japanese high command.

That is, because the announcement of the bombing was delayed by 16 hours, and because the Japanese (understandably) wanted to have a scientific team confirm that it was an atomic bomb before making any decisions based upon it, there was simply not enough time for them to do that and hold a formal meeting about it.

But again, this entire framing completely misses what the goals of the American planners were — they were not, at all, concerned about saving Japanese lives, about not using the atomic bomb, and they did not expect the Japanese to surrender immediately after the first bomb. The entire idea that the use of the bomb was something done regretfully or "only as a last resort" or anything like that is another after the fact narrative created to make it seem justified and necessary, and in no way sums up how any US policymakers looked at it at the time — they had a new weapon for war, they fully expected to use it.

(For a lot on the decision-making that took place on Tinian, see Michael Gordin, Five Days in August. For the Japanese high command's side of things, see Tsuyoshi Hasegawa, Racing the Enemy.)