I read your question carefully and I understand your concern.
Actually, I need more information regarding your problem, for example how many days have passed since your big toe trauma, is there a blue/black discoloration of the toe, have you noticed any pus around or beneath the toe nail, is your big toe all red and warm?
I will try to include all possible options. Since it's a trauma, I don't think that you have an
ingrown toenail (unguis incarnatus). The pain you have is due to trauma of the toenail itself and the surrounding tissue by the toenail. You can take over the counter pain killers but I don't think it will help to much. In order to get the best management you can consult your primary doctor, so he/she can see objectively your issue, since the treatments include from simple local wound care to removal (partial or total) of your toenail, antibiotics and regular local wound dressing (it depends on severity of trauma, weather it is infected or not). Your doctor also have to visit you objectively, to rule out also a possible bone trauma of your big toe (but in the way you describe the trauma, I don't think that's your situation)
I'm at your disposal if you have additional questions. I hope my answer helped. Wishing you a good health
If you can, in order for me to give you the best advice, is to send me a picture of your traumatized toe.