Answer: Leyland cypress is often recommended for a fast growing hedge, but it does prefer full sun all day and will require frequent trimming to maintain that six foot height because it is naturally a much larger plant.
Privet, although technically deciduous, holds its leaves late in the season and leafs out early. It is eminently trimmable and will tolerate partial shade, although it too preferes full sun.
A Japanese holly such as "Skypencil" would do well in partial sun and is very narrow in its natural form. This might be a plant to consider.
The Euonymus "Manhattan" might be another to consider, although it may not be quite as tall as six feet for you, it will do well in partial shade and is an evergreen.
Another possibility for quicker coverage is to install a fence and soften the look of it with a vine such as the evergreen Hedera helix (English ivy) or perhaps a blooming plant such as the fragrant, white flowered fall blooming clematis which does very well in partial shade.
I hope this gives you some ideas. Your local area nursery personnel might also have some suggestions based on additional information about the growing conditions in your yard.
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