The short answer is no.
In space the rocket exhaust isn't really a gas it's more like a molecular dust, where all of the molecules (CO, CO2, H2O, etc) are travelling on their own divergent trajectories. In order for a 'gas' pocket to form those molecules would need to be held together in some way, and there's just nothing acting on them to do this. Instead you have a variety of factors acting to spread them apart, including their own individual momentum.
And don't forget: Space is big. Really big.
This document gives details on one LEO flight of the Falcon 9S9 (Falcon Heavy). The second stage fuel load was 107,500 kg of fuel (LOX + RP-1). 100.5 seconds of return burn accounts for the bulk of the fuel, 15.6 seconds for the landing burn used most of the remainder. Approximately half of the return burn took place above the Karman Line, and from the acceleration curve it looks like it accounts for about 45% of the expended fuel. Let's say that's on the order of 50,000 kg of combustion products spread over a couple of hundred km of trajectory in under a minute.
From the picture in Hobbes's answer you can see that the exhaust spreads out pretty quickly, and with very little to impede it that spread is going to continue. You'll end up with a curved plume of molecular 'dust', most of which will either fall back into the Earth's atmosphere or escape orbit on its own momentum. At low orbits there are enough other gas particles that most of the exhaust probably won't make it out of the magnetosphere to be swept away by the solar wind.
So yes there is a slight increase in the density of matter in low orbit, but it's extremely thin and relatively short lived. It would take some hypersensitive apparatus to detect it at all by the time of the next launch.
On the other hand, a handful of molecules may be ejected with just the right momentum to enter an elliptical orbit around the Earth. Come back in a few centuries (if we're still using combustion to get to orbit) and you might notice an effect. At worst it is probably going to require more orbital corrections per year for LEO satellites, which already have problems with the density of molecules so close to Earth's atmosphere.