Lots of fun things about that. But presuming for a moment that we actually could change the earth's orbit but pushing something into the ground, doing so at mid-day wouldn't move us any farther from the sun.
Pretend that you have an earth-pushing device and you're standing on the equator and the sun is directly overhead (doesn't happen every day). If you push the earth outward, away from the sun, what you're effectively doing is fighting the sun's gravity right? The sun is pulling inward and you're pushing outward. It's essentially like the sun lost a little bit of gravity.
So what would happen if the sun suddenly lost a little bit of gravity?
Our orbit would slow down. It doesn't move out or in, it just slows. It takes longer to get around the sun once. So pushing against the earth won't actually move you outward (unless you have enough force to escape the sun's gravity).
To increase the earth's orbital distance you need to push on the terminator (sunset). But if you do that once, you're actually only increasing the orbital distance on the other side of the sun, when you get back to the time when you pushed you'll be right at the same distance you were. So you need 2 pushes to increase the orbital radius but maintain a circular orbit (or low eccentricity). One push on one side of the sun (at sunset), and another equal push on the exact other side of the sun (at sunset). This will increase the orbital radius while keeping it circular.
If you don't do it at the right latitude, such that you're pushing exactly into the Earth's orbital plane (this will depend on the time of year), you'll end up changing the orbital plane itself. This is known as the inclination of the orbit. This wouldn't necessarily change the distance from the sun, but it would require you to use extra force since some of the force would be wasted on changing the orbital plane instead of boosting the orbital radius.
Technically, you also need to perform a maneuver on the moon as well. What you're really trying to accelerate is the Earth-Moon barycenter. Unfortunately I think you're guaranteed to screw up the Moon's orbit a bit when you do that.