Smee's famous WB diagram giving shear forces.
So, the methodical way is to compute the horizontal shear, divide by the cosine of the angle of the cord, and voila! A whole lot of force on that stake.
That's right. Increasing the line reduces theta and increases cos(theta).Also, for the stakes, making the line longer (i.e. staking down farther away from the hammock) would be an easy way to reduce the amount of "pull out" force. The stake is going to hold better if there's less upwards and more sideways force acting on it.
One more thing you can do is use multiple lines and stakes to share the load. You want them all pretty close to each other because the wider the angle of the cord from center, the less effective it is at giving opposing force in the direction that needs opposition.