I did exactly that and it doesn't do anything.
Supposedly to fight him again you
1. Talkt to receptionist
2. Talk to his mom
3. Mom talks to Jackson
and then you can fight him again.
But for me, it doesn't work, no matter the order or how many tries.
I just started out in Kanto and I lost the 2nd fight to Jackson in the Viridian City School.
Now the NPC in the underground doesn't let me go back in and when I talk to the receptionist, she just says "Hello..." so I can't talk to his mom after to get her to let me fight Jackson again.
I have tried relogging multiple times and talking to the NPCs in different orders.