One problem with the owls is that Britons usually want to see them, hearing the call is not good enough. Some owls call spontaneously in autumn, especially Pygmy Owls very early in the mornings (just before and at dawn) in August and September. However, Pygmy Owl is pretty scarce in Lapland, it's much commoner in e.g. southern Finland.
Usually young Owls move around a bit in the autumn, and some years there are influxes of especially Pygmy and Hawk Owls (some Hawk Owls stay at the same location for the duration of the winter, which makes them easy to twitch). Hawk Owl is one of the easier owl species to see, it's active during the day also and often perches quite openly (it breeds generally only in northern Finland). Also some Tengmalm's Owls migrate in the autumn, but they are much more difficult to find than Hawk and Pygmy Owls.
http://www.tarsiger.com/gallery/index.php?pic_id=petro1232383695&lang=eng
Hearing calling owls is easier in March, but it's not overly easy then either (depending on how much voles there are around), and seeing them is difficult, unless you know the exact breeding locations. However, the summer birds have not arrived in March, so most foreigners visit Finland in early June when you have a chance of seeing all the species (the owls usually with the assistance of a guide who knows the locations of the nests).
Like Jos wrote, Siberian Jay, Siberian Tit, Pine Grosbeak and the woodpeckers are completely possible to find in the autumn, too. During the last few years there have been some Red-flanked Bluetails at the nesting sites in Kuusamo still in the beginning of September (but September is not the best month to visit Finland if you want to see one). Black Woodpecker is rather easy to find at any time of the year (in entire Finland except the on the treeless tundra), Three-toed Woodpecker is a bit more difficult (it's a bit more common in the north, but like Jos wrote, some years there are influxes of eastern birds). White-backed Woodpecker doesn't occur in Lapland and Grey-headed is pretty scarce there (most records are actually from the autumn and winter).
Nutcracker is relatively easy to find in southern Finland in autumn (if you know some good locations), but more difficult in Lapland (it's still much easier to find there in autumn than in summer because of migrating birds from Russia).
It is possible to see Rustic Buntings in north-eastern and eastern Finland in the beginning of September (or mid-September), it's more scarce in in western Finland, as they migrate towards south east.
Hazel Hen is completely findable in autumn, for example the hunters shoot about 90.000 of them each autumn. They respond quite acively to tape in autumn.