The Soviet Union had been planning for the war for quite a while. Already in 1938 - that is prior to the Molotov-Ribbentrop Pact (MRP) - Soviet war games in the Leningrad district were based on 'aggressive defense' after a 'theoretical' border incident at Mainila. So they had been preparing for the war quite some time by the end of the 1939. Most historians think that the ultimate Soviet goal was to conquer Finland and either turn it into a subject or outright annex it (opinions vary on that) in similar manner like what the Soviet Union did to Estonia, Latvia and Lithuania. Moving the borders was just an excuse just like the base & troop agreements were with the Baltic states. Among other things the Soviet creation of puppet state (Finnish Democratic Republic) and subsequent refusal to recognize the actual Finnish government, Soviet plans of capturing the Finnish capital, captured Soviet orders prohibiting the Soviet units from crossing into Sweden (on the opposite corner of Finland from the contested area) show this.

As why to attack late in the 1939... Let's start with political side. First you need to realize that the picture Soviets got from Finland was vastly different from reality. What i mean by this is what the Soviet spies reported back got twisted and turned as it went through the Soviet bureaucracy to give an 'acceptable' view of Finland to the Soviet leaders. In other words aspects not supporting the Soviets were redacted and those even slightly in favor of Soviets were highlighted. Then you have the MRP which gave Soviets freedom of operations against Finland while the Western Allies had been shown to be unable to take any meaningful action. So there was little chance of effective support for Finland, Germans were still facing & holding down the Western Allies, and Finland was reported to be unstable.

Then the other aspects... Finland had very few land routes leading out of the country with any sort of actual capacity so in a logistical sense it could be seen somewhat like an island. This meant that ships were needed to transfer any meaningful quantity of material into Finland. However during the winter the northern Baltic Sea would freeze up helping to isolate Finland. Additionally the Finnish terrain is filled with swamps, marshes, rivers, streams and lakes. During the winter those could be traversed with relative ease especially by the infantry but they would be a huge problem especially in the autumn and spring. This is because of the 'kelirikko' or 'rasputitsa' which would turn the roads (there were very few if any paved roads in the area) along with other terrain into quagmires and turn the previously passable rivers and lakes into obstacles. While some military vehicles and infantry could traverse such terrain the logistics rarely could which would make any practical attacks in those conditions very difficult to carry out. So the only reasonable times for launching an invasion would be winter or summer.

Just to be clear, there wasn't any kind of military threat, time sensitive or not, from the side of Finland in that region in 1939 or in the decade preceding it.