I go to my assigned grid and at the very least complete the 24 hour patrol there. I added the renown for doing that back into my game, in large part because it rewards me for resisting the temptation to ignore my "orders" and just go where I think the hunting will be good.
After I've completed the patrol assignment, it's kind of a toss-up as to what I'll do next. Generally I will stay in that area, but expand my patrol area into adjacent grids, unless I get reports of enemy activity that I have some chance of intercepting in which case I'll go where the expected action is.
If I don't get any reports that direct me towards potential action, I'll try to work my way through the grids adjacent to my original assigned area in such a way that I'm moving towards a known shipping lane if at all possible.
IIRC once I've had some success and sent in a patrol report, the reply I get says something about continuing to patrol "at [my] discretion" as long as the fuel and weapons hold out. So I tend to take that as the freedom to go where I please, within reason - if I was sent to patrol the upper Western Approaches, for instance, I'll probably continue patrolling the known shipping lanes west of Ireland rather than setting course for Gibraltar.
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