Your residual speed at the surface needs to be the escape velocity from the surface.
But you can try using conservation of energy.
... assuming constant density for the Earth.
Basically you want to know what initial speed you have to be doing to escape a mass distribution - starting from within
the distribution. This sort of thing is pretty hand-wavy since, for something like the solar-system, the specific path taken is important (i.e. if all the planets are on the other side of the Sun to the Earth when you start, you can start out slower maybe.)
So the concept of escape velocity is a bit vague in this scale.
Using: hyperphysics solar system data
From the Earth/Moon orbit, you need 42.1kmps to escape the Sun's gravity.
If you used the entire
mass of the solar system you'd get something like a 0.04% difference to this.
You need 11.2kmps to escape the surface of the Earth - but still bound to the Sun.
To escape the Sun as well, you need 42.1kmps residual speed ... so, <waves hands around> 53.3kmps would get you out of the solar system from anywhere bound to the Earth.
If you get too precise, you have to take the planetary dynamics into account.
Anything else is an approximation.