Let $N \ge 5$ be a prime number and $E/ \mathbb{Q}_N$ be an elliptic curve with additive reduction. Then it is easy to see that there exists a finite extension $K$ over which $E$ has stable reduction.
I want to show that we can choose $K$ so that the ramification index of $K/ \mathbb{Q}_N \le 6$.
The proof of VII 5.5 of Silverman’s AEC does not include that.
This minimal ramification index is the order of the Serre-Tate group $\Phi$, defined in their article "Good reduction of abelian varieties". It is shown in the proof of theorem 2 there that $\Phi$ is a subgroup of the automorphism group of the reduced elliptic curve over the larger field. If the residual characteristic is not 2 or 3 then the automorphism group is cyclic of order 2, 4 or 6. This gives you what you wanted. In fact $\Phi$ is cyclic of order 2 if the reduction type is I*${}_n$, it is cyclic of order 3 for type IV and IV*, cyclic of order 4 for type III and III* and cyclic of order 6 for type II and II*. For residual characteristic 2 or 3 it is all far more complicated.
