Personally I have the Image of Lavellan one day leaving the inquisition behind, putting her advisers in charge of holding things together while she's gone looking for Solas. Of course Dorian, Cole and Blackwall would go with her while the rest of companions split up bouncing between following leads Lavellan's group are far away from, and helping the Inquisition uphold its duties without her. Any information that is found would be given to Leliana who would sort through it and verify it before sending scouts to deliver the information to Lavellan.
As they traveled they'd do good deeds for the people, Cole would point out someone who needs help and they would help asking only for them to keep the look out for a bald elf wearing a wolf jaw necklace in return and perhaps the occasional supplies that they've run low on from people more fortunate than those who need help. Dorian would constantly be trying to keep our inquisitor cheerful, with winks and a little play flirtation, some days even helping her finish a bottle or two of liquor before she fell asleep at camp. Blackwall would feel at home in the wilderness, using his warden status to help them get small supplies when weapons break or become dull.
I feel all warm and fuzzy thinking of all the bonding they'd do. Dorian and Lavellan would pass ideas about magic, the fade and history back and forth while Cole would chime in with what he thinks Solas would say, or what he himself thinks. I could see Blackwall bonding with Cole in a fatherly sort of way, taking him out to help him find food while Dorian and Lavellan talk or research things they've found. Often the four would find themselves with odds against them, standing back to back defending themselves from Undead, Demons and Bandits. Lavellan would learn a few small healing magics, nothing major but enough that she can help patch up cuts and bruises back at camp when things settle.
They'd piece together lore than the Dalish only dreamed of knowing, delving into ruins that keepers would never let the hunters near. Perhaps they would even begin looking into the other Magisters that had entered the black city, looking for a connection between them, the old gods and the Dalish gods. The potential existence of other beings like Corypheous would be the Inquisition's official statement on why it's leader is absent as they would claim she is someone who possesses great knowledge and is best suited for this kind of searching.