1996 Toyota Celica Smoke coming out of the tailpipe and bur

Engine Performance problem
1996 Toyota Celica 4 cyl Manual 280000 miles

I have a 1996 Toyota Celica with 280k miles. Recently while filling the oil, I forgot to screw back on the oil cap and this caused the oil to leak everywhere. I finally had the engine washed and most of the oil was washed out. Everyday when I go to work about 15 minutes into my driving I smell oil burning and at every stop I see smoke coming out of the hood but the car runs fine and the temperature stays normal. I figure it's just burning the oil that's left. Today, about 25 mins into the driving a large amount of smoke came out of the tailpipe and when I got to work and opened the hood a lot of smoke was coming out from underneath the car. When I turned on the car, my friend told me that he saw oil coming out of the tailpipe and a spray of oil, but aside from that the car runs fine. What could be causing this?