Indeed, leaving the village to do something like this was quite displeasing for any village leader. However, Jiraiya had spent the past several years working tirelessly for the village, had trained a genius like Minato Namikaze, and had even formally signed a peace treaty with Sunagakure earlier.
It could be said that he had already done more than enough. Now that he wanted to go to Amegakure to teach Ninjutsu to his three students, even if the Third Hokage was a hundred times unwilling, there was nothing he could do. Besides, the Third Hokage understood that even if he stopped Jiraiya now, Jiraiya would sneak off on his own anyway. It was better to agree directly than let things turn unpleasant later.
Very quickly, a month passed. Jiraiya packed his things and prepared to leave for the Land of Water, while Li did not think there was anything wrong with it. After all, according to the original plot, Jiraiya should have remained in the Land of Water after fighting Hanzo of the Salamander to teach Yahiko and the other two Ninjutsu.
Now, however, Konoha's higher-ups had forcibly assigned him a cheap disciple like Li, leaving him no choice but to return to Konoha. This could be considered a change to the original plot.
After Jiraiya left, though, Orochimaru had no time for him instead. It seemed Orochimaru was fully occupied researching the forbidden scroll from Kumogakure that Li had given him. He had probably not thought much of it at first, since Konoha itself had collected forbidden scrolls from Kumogakure. But after examining it carefully, he should have realized that Li's copy was far more complete than the transcripts previously kept by Konoha and the various clans.
There was no news from the front lines during this time. By the