I've designed this circuit with 4 logic gates, and did Karnaugh map's simplification and Quine McCluskey method. However I found out that actually my circuit design is already optimized and I can't really compare how the simplifications offer a less expensive circuit.
I'd like to add a few more gates to the circuit not changing the boolean function of it. Which is (X2'∙X0)+(X2∙X1)
You are correct in your desire to add extra gates, because as it's currently constructed it is prone to a logical hazard. Particularly there is a risk of the output briefly transitioning to a $0$ when the input changes between $111$ and $011$ when it should remain $1$.
Adding the term $X_0X_1$ will not change the logical function of the circuit, and will eliminate the hazard. So your expression will become:
$$F=X_0X_2'+X_1X_2+X_0X_1$$