Sweet Alyssum (flower) is classed semi hardy in the cited extension guidance. The timing rules below are relative to your local freeze dates — enter a location to turn them into calendar dates.
The rules this site applies
Method
Frost-relative window
Rule & reconciliation
Start seeds indoors
6–4 weeks before the median last spring freeze
Colorado State's seed-starting table lists sweet alyssum (Lobularia) at 6 weeks before the average frost-free date in the 'Indoors' column, the only indoor figure found that is anchored to frost rather than to the outdoor planting date, so indoor sowing opens 6 weeks before the median last freeze.
Transplant outdoors
2–1 weeks before the median last spring freeze
Wisconsin Horticulture puts hardened-off transplants in the garden a week or two before the average last frost, which starts later than Illinois' four-weeks-before hardy-annual rule, so the more frost-conservative Wisconsin interval governs.
Direct-sow outdoors
4–2 weeks before the median last spring freeze
Illinois Extension names sweet alyssum in its hardy-annual list and quantifies those at about four weeks before the frost-free date, and Clemson corroborates the direction by saying seed should be directly sown several weeks before the last frost date, so direct sowing opens 4 weeks before the median last freeze.
Variety assumption: Assumes seed-grown Lobularia maritima, which Clemson notes flowers within 6 to 8 weeks of sowing and may stop blooming as summer heat rises; sources disagree on hardiness — Illinois and NC State call it a hardy annual while Clemson lists it as half-hardy, so the more cautious semi-hardy class is used.
"Half-hardy annuals – such as allysum and dianthus -can tolerate light frost and are usually planted in early spring for spring and early summer color."
"Transplants that have been hardened off can be placed in the garden a week or two before the average last frost, otherwise tender transplants should be protected until add danger of frost is past."