The pressure in the plasma in the solar corona increases outwards from the Sun, and the corona does not end near the Sun but continues out into interplanetary space. The result of an increasing coronal pressure and a decreasing gravitational field strength AND a decreasing density of particles is an increase in the wind speed away from the Sun. Simple models show that the wind speed increases rapidly from 'zero' to 350 kilometers/sec within 20 million kilometers from the Sun for a coronal temperature near 1 million degrees. By the Earth's orbit, this speed has increased to about 400 kilometers/sec, and by the orbit of Uranus it achieves about 450 kilometers/sec...and is in fact traveling at Mach 25 because the temperature of the gas in the wind declines with temperature so its sound speed drops quickly outside the Earth's orbit.
Think of this as the Sun putting energy into the wind as a whole, and to conserve energy wile the density of particles decreases away from the Sun, the average particle has to travel at increasing speeds...just like the cracking of a whip. The tip moves a lot faster than your hand that 'jiggled' the whip. But the energy has to be conserved as the wave travels down the narrowing whip taper.