They reach a breakthrough with the experience-syncing with the clones eventually, though it's mildly distracting. They solve this by upgrading the clones so they have much more autonomy, so they can choose to alert Tetsuma to problems. (An off-branch of this is the Transmission Jutsu - Tetsuma can turn a clone into a seed, then attach it to something and track that something's location, though they can't get any other feedback while the clone's transformed.)
They also figure out the merge-with-wood jutsu! It's ridiculous. Completely ridiculous. Tetsuma can't manage it, but their clones can merge with a tree or the ground and travel quickly through the network of underground organic matter. The submerged clones are undetectable to Tetsuma's chakra-sensing abilities, only slightly detectable to the wood-sense, though they have no idea if anyone in all the realms would be able to detect them. They name it the Mayfly Jutsu.
...They also figure out how to make wooden spikes launch out of their body, because why not. (In the time before the feast day Tetsuma might get slightly bored while Dayo's away.) More usefully, they improve the making-shelter jutsu, and create a nice little house in a clearing in the woods (they were getting tired of being rained on), with mild genjutsu around it to misdirect anyone who wanders by.
They don't really find out much during their excursions into town, or from their from-a-distance spying on the castle, other than general information about when the castle's emptiest, and that the people in this village over here really don't like the lord (they're mostly weavers, and the tax on cloth is Unfair. Apparently there was a decades-old dispute between the weaver's guild and the lord, where they figure the higher cloth tax came from).