In this paper, we consider the perturbed KdV equation with Fourier multiplier $$\begin{aligned} u_{t} =- u_{xxx} + \big (M_{\xi }u+u^3 \big )_{x},\quad u(t,x+2\pi )=u(t,x),\quad \int _0^{2\pi }u(t,x)dx=0, \end{aligned}$$ u t = - u x x x + ( M ξ u + u 3 ) x , u ( t , x + 2 π ) = u ( t , x ) , ∫ 0 2 π u ( t , x ) d x = 0 , with analytic data of size $$\varepsilon $$ ε . We prove that the equation admits a Whitney smooth family of small amplitude, real analytic quasi-periodic solutions with $$\tilde{J}$$ J ~ Diophantine frequencies, where the order of $$\tilde{J}$$ J ~ is $$O(\frac{1}{\varepsilon })$$ O ( 1 ε ) . The proof is based on a conserved quantity $$\int _0^{2\pi } u^2 dx$$ ∫ 0 2 π u 2 d x , Töplitz–Lipschitz property and an abstract infinite dimensional KAM theorem. By taking advantage of the conserved quantity $$\int _0^{2\pi } u^2 dx$$ ∫ 0 2 π u 2 d x and Töplitz–Lipschitz property, our normal form part is independent of angle variables in spite of the unbounded perturbation.