A discone is a bad idea since it will pick up more out of band signal.
Most of these 1090 antennas are based on cookie cutter designs scaled from the ARRL handbook. With one exception, I never found one that was modeled in NEC2.
These colinears look good on paper. Trouble is they will have nulls in the reception and these will not be found logging your most distant hits. In my opinion for what is commercially available, I would get a simple half wave monopole and call it a day.
Note that no monopole will beam to the horizon let alone beneath it. I have a folded dipole design fully modeled in NEC2. I find the impedance match to be troublesome. The quarter wave didn’t work at all. I have a exponential taper match kind of working but it looks too bulky. I will try a Klopfenstein match, but the math looks ugly.
The 1090 I saw NEC2 modeled was based on
This will beam to the horizon but it is directional. Of course given the cost of dongles these day it might justify have multiple antennas/receivers if you are gonzo enough.