Alyssum is popular because of its lightly fragrant and lacy white bloom that lasts through most of the year. It seems to be more perennial than it actually is because it sows seed to replace aging plants. Candytuft, Iberis sempervirens, is a bit less prolific with bloom and fragrance, but otherwise resembles alyssum. Without seeding, it can be nicely perennial.
Candytuft does not get much larger than alyssum although it supposedly has potential to get almost a foot high and a foot and a half wide. Shearing after bloom phases enhances foliar density and subsequent bloom. Primary bloom occurs during late winter, spring, or perhaps early summer. Minor random bloom is possible at any time, particularly autumn.
Plants propagate readily by division of small tufts of rooted stems from within established plants. Alternatively, creeping outer stems develop roots if simply pressed into the soil or held down with stones. Pruning scraps are tiny and awkward to handle, but can grow as cuttings. When disturbed, candytuft exudes an aroma similar to that of related cabbages, which might be objectionable to some.
Tony Tomeo 
Candytuft is one of my all-time favorites. I have a couple of places where it’s been for twenty years but efforts to establish it other spots around the garden have been unsuccessful. I’ll try your techniques for spreading it.
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I prefer layering to cuttings. Cuttings take a bit of attention. I can do the layering instead of cutting back the squiggly growth. It does not take much effort at all. I literally press a bunch of it into the soil and come back later to take it away. Division is better than cutting, but not quite as rewarding as layering, and can leave a temporary bald spot.
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I’m a huge fan of Iberis ‘Snow Cone’ which edges our front walkway. I shear it lightly after the first blast of bloom is over, just to keep that nice dark green without the distraction of aging flowers. We normally see a light repeat bloom in the autumn (Sept-Oct) here as well. I even had a few putting out flowers in December 2019! Not sure whether they were very late autumn ones or super-early spring ones, lol
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Actually, although I prefer to shear ours back in winter, after the autumn bloom phase (instead of after the spring bloom phase), I can shear it back whenever it gets shabby, at any time of year. It sort of blooms whenever it wants to anyway. I suppose that I should shear it after spring bloom, so that it can regenerate faster, but I dislike how it looks after the redwood debris gets raked off of it after winter storms. If it is a cultivar, I do not know. We have been plucking and plugging the same plants for quite a while.
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