Consider the homogeneous diophantine equation $$ax_1+bx_2+cx_3=0$$ over $\mathbb{Z}$ with a, b, c coprime.
(A version of) Siegel's Lemma states, that there exists a non-trivial solution $x$, such that $$\max(|x_i|) \leq \sqrt{(a^2+b^2+c^2)^{\frac{1}{2}}}. $$ Another version states, that there exist two non-trivial linear independent solutions $x, y$, such that $$\max(|x_i|) \max(|y_i|) \leq \sqrt{(a^2+b^2+c^2)}. $$
Now suppose I already have a solution $x$, which satisfies the first inequality. Can I infer the existence of a second solution and bound it using the second inequality with the $\max(|x_i|)$ value of the first solution?
So perhaps this is a counterexample you are looking for?
Take the equation $$4x_1 + 5x_2 + 9x_3 = 0.$$ We have a solution $x = (3,3,-3)$ which satisfies $\max|x_i| \leq 122^{1/4} \approx 3.68$.
The next linearly independent solution of smallest height seems to be $y = (6,-3,-1)$, but $$\max|x_i| \max|y_i| = 3 \cdot 6 = 18 > \sqrt{122} \approx 11.04.$$