Lilies only bloom once in the season, not continuous bloom. We lily growers stretch our season by buying lilies from their different divisions. Asiatic and LAs (Longiflorum x Asiatic) generally start in June followed by trumpets, Orienpets (Oriental x trumpet) and Oriental lilies. Some of each of these groups give bloom June through at least August, and sometimes into September.
Continue caring for your Asiatic lily. Don't trim the green leaves off as they help prepare the bulb for next year's bloom. Set the container aside in a shady or partly shady place and give a little water from time to time so that the bulb doesn't completely dry out.
If you do want to stretch your season pick up and plant some new bulbs in the fall from the different groups and plant them then.
