Can't help with Manitoba... but when I lived in New Brunswick and didn't have a dog, it was possible to bumble around in the alders or poplars and occasionally flush a few. But I got most woodcock by going out to old gravel pits just before dusk, and waiting. It helps if the surrounding cover includes alder or aspen/poplar, but it isn't required. I know, it sounds like snipe hunting with a sack & flashlight, but I swear it works! I didn't believe it at first, but over the years I got quite a pile of woodcock this way. The advantage of being in the old pits is that you can hear them coming (wings make a distinctive whistle) and see them in the open. Woodcock actually fly pretty slowly. I remember missing one with both barrels, then reloading and getting it on the 3rd shot!
Otherwise, go with someone who has a good dog. Again, alder or aspen cover is best.