To me this is the best time for the players to consolidate gains, create networks or reacquaint themselves with family, old friends, allies, etc, to advance financially and economically, to collect rewards and pay for service, get promoted, to train, build fortifications and keeps and houses, advance in level and to advance socially, politically, and professionally.
So to me and to them it's like a campaign within itself, but the goals are different than they are for missions, scenarios, and adventures. With missions, scenarios, and adventures, the players are usually working for some aim or objective over which they only have limited contour against forces and enemies which are generally unknown, and usually in the employ (direct or indirect) of others.
Between Campaigning though (and it is called campaigning since they are a paramilitary and intelligence outfit working for the Byzantine government and the Orthodox church) the players have a chance to undertake their own desired goals, to collect rewards and pay, to distribute and investigate treasure, and to pursue hobbies and personal interests and other ends (such as building, investing, getting married, raising kids or livestock or both, sailing, advancing socially, and so forth) over which they usually have direct and personal control. In addition rather than dealing with enemies (although occasionally enemies will track or harass them, both inside Constantinople and at their own estates) they can build networks of friends, family, and allies. They can also invent, prepare for future campaigns, read, study, train, recuperate and convalesce, get treated, pursue religious devotion, travel, vacation, etc.
Oftentimes the periods between Campaigning (how much time is spent between one campaign and another) are determined by need, season, or the orders of superiors and the Emperor. If not then generally I let the players decide how much time they spend between Campaigns and what they do in that time.
Time spent not Campaigning is called Relief, or being Relieved.
We've had a lot of good adventures or preparations for new missions and adventures during relief time.
Vacation, travel, training, vadding, study, business, building, farming, hunting trips can all make for good adventures during Relief. Usually no one is trying to kill them but there are many other dangers, opportunities, and enterprises available to people aside from combat. That's as true of the imagination as it is in real life.