There are basically three versions of 24" wheel:
ISO 507 - used on kids' bikes and mountain bikes. This has the wider tires available.
ISO 520 - 'road' version of 24" generally limited to 1 1/8"
ISO 540 - wheel chairs. Narrow only - good luck finding anything other than 1".

All are rare in recumbents. Your fractional sizing implies either 520 or 540. Read your sidewall to be sure. If it's 520, the fattest tires I know of that are available are the Terry Tellus 24x1 1/4 or the Vee Rubber 24x1 3/8 Micro Knobby. With the 507 size, the big trouble might be finding a tire as narrow as 1 3/8".