The trace was from the output of the LM-358, and the hall sensor was a UGN3503U, but i think any linear hall sensor should work.
I dont see why you have a led across the coil as at 3khz do you think you will see it flash??????
I think the supply voltage might be too high as the coil will be too powerfull, requiring the magnet to be further away, then you have the problem of the magnet being too far away from the hall sensor to read, so you endup in a loop of things not working, try 5 volt supply.
The reason i post schematics is for all to use and to share the information, so of course you can use it.
I really dont care what chips you use, for your final project, my only aim was to show how simple this can be to get it to work, as all those who try a fancy design come here asking why it wont work.
As i said i too tried a micro and other design ideas, but the best result was back to basics and keep it simple.
Pete.