They most certainly advanced past the pre-Winter War border. On the Karelian Isthmus, they only advanced a little past the old border, to obtain a fairly straight defensive line while recapturing all of their pre-Winter War territory. The advanced further in East Karelia, reaching Lake Onega, up to the Svir River joining the southern ends on Lake Ladoga and Lake Onega. North of Lake Onega, they advanced to approximately halfway between the border and the White Sea. Further north, in Lapland, was primarily a German theatre, and the German attempt to advance and capture Murmansk made little progress.

Two maps that show the pre-Winter War and pre-Continuation War borders, and the extent of the Finnish (and German) advance:

• https://commons.wikimedia.org/wiki/File:Continuation_War_1942_and_Soviet_assaults_English.jpg

• https://commons.wikimedia.org/wiki/File:Finland_Administrative_map_1942_1944.png

Why advance past the old border? The Finnish attack was not just an opportunistic attempt to recapture their old territory alongside the German attack (Operation Barbarossa). It was planned in advance together with the Germans, and Germany certainly wanted more than just Finland regaining its former territory. If Germany defeated the Soviet Union, Finland would gain East Karelia (and perhaps more - the largest goals of those who sought Suursuomi, "Greater Finland", included the Kola Peninsula, Ingria, and Estonia). If Germany lost, Finland would have space to retreat before having to fight on Finnish territory, and could also offer to retreat to the pre-Winter War border in negotiations with the Soviet Union.

What disadvantages resulted from advancing past the old border? First, the Western Allies (notably, the UK and USA) threatened to declare war on Finland if Finland didn't withdraw to the old border. In particular, the advance towards the White Sea threatened the Murmansk railway. Finland's response was to stop the advance, but not return to the old border. The Allied response was that Britain declared war, but didn't attack Finland (apart from bombing mostly German shipping in Petsamo on the Arctic coast), and the USA didn't declare war on Finland.

After the Finnish advance halted (after about 4 months of fighting), and the front became relatively quiet until the Soviet offensive of June 1944, which broke the Finnish lines, and pushed the Finns back into Finland, where after some further Soviet attacks were halted, a ceasefire was agree in early September.

This was not the end of Finland's involvement in the war - the Continuation War had ended, but the Lapland War had begun, with Finland fighting to eject German forces from Lapland. This fighting continued from mid-September 1944 to the end of April 1945. It also involved a German amphibious landing to capture the island of Suursaari (Operation Tanne Ost); this attack failed and about half the German force was killed, wounded, or captured (mostly captured).

How did Finland contribute to the war overall? The Finnish front tied up about 500,000 Soviet troops, and the Soviets suffered almost 1 million casualties on the Finnish front (about 1/3 killed or captured, and the others wounded and sick). This was not insignificant. The number of German troops peaked at over 200,000, with the Germans suffering over 80,000 casualties (a little over 1/3 of the Finnish casualties of over 200,000).

Among officers (and chaplain) there was often enthusiasm for the conquest of Greater Finland. This was not as common among the common soldiers, who were willing to fight to defend their country, but sometimes less willing to fight to conquer somebody else's country. This was noted in the famous novel of the Continuation War, Väinö Linna's Tuntematon Sotilas, "Unknown Soldier(s)":

‘Guys, the old border.’

The event revived their spirits somewhat. Hietanen, standing in the middle of the clearing, took one big leap and said, ‘Aaand now! The Hietanen boy stands on foreign soil!’

‘We’re in Russia now, boys,’ Salo said.

Lahtinen hobbled over irritably, glaring at the others out of the corner of his eye and muttering, ‘So we are. And here our rights end. By which I mean, from this point on, we’re a pack of bandits. Just so you know.’