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Suppose I have a G2++ short rate model: $$r(t)=x(t)+y(t)+\phi(t), \quad r(0)=r_0$$ with $$dx(t)=-ax(t)dt+\sigma dW_1(t), \quad x(0)=0$$ $$dy(t)=-bx(t)dt+\eta dW_2(t), \quad y(0)=0$$ $$d\langle W_1,W_2\rangle_t = \rho dt$$ Although it has many features that are typical stylized facts of the yield curve, the does not seem to capture well the volatility smile/skew. I would like to know whether there are some model extensions enabling to partly capture the smile/skew, without switching to a different model, such as Cheyette, SABR or ZABR.

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