Let's say that there is a solution
$\ce{Na+(aq) + Cl-(aq) <=> NaCl(s)}$
If the above solution is at equilibrium we can say that it is saturated and hence no more Sodium Chloride will dissolve. However, according to LeChatelier's Principle, if we add more Sodium Chloride it will just dissolve. Is there another rule that governs how this kind of dissolution behaves?
You misinterpret the principle. It does not say if you add more salt, more salt dissolves. More generally, the principle focuses on changes of intensive properties, that do not change with the system scaling.
The principle says if the thermodynamic activity of the solid had increased by adding more salt, more salt would have dissolved to decrease this activity. But by adding more salt, the salt activity remains constant.
You can also consider chemical potential $\mu = \frac{dG}{dn}$ instead of activity, as they are closely related.
By other words, thinking more salt dissolves by adding more salt to saturated solution is equally wrong as thinking more salt crystalizes by adding more of saturated solution. As the salt activity in saturated solution or in solid state does not depend on their amount, which is extensive property.
