Are K_t-minor free graphs on small vertex sets understood?

In a paper on Hadwiger's conjecture, https://web.math.princeton.edu/~pds/papers/hadwiger/paper.pdf, Seymour explains various results on excluding the complete graph as a minor.

In particular, there is a nice bound on the number of edges, due to Mader, as follows:

If $$t\leq 7$$ and $$n\geq t-2$$ then every $$K_t$$-minor free graph $$G$$ on $$n$$ vertices has $$|E(G)|\leq (t-2)n-\frac{(t-1)(t-2)}{2}.$$

(The obvious application to Hadwiger, as Seymour describes, is that this bound controls the average degree and hence gives low minimum degree to which you can use induction to get a colouring result.)

Sadly the pattern fails for $$t\geq 8$$. Jorgensen and then Song and Thomas describe the exceptions explicitly for $$t=8$$ and $$t=9$$ respectively but as far as I can see no larger $$t$$ is understood. To be more precise Jorgensen's result says that if there is no $$K_8$$-minor and the displayed inequality fails then the exact number of edges is known and $$G$$ can be built from $$K_{2,2,2,2,2}$$ in a simple way (by "pure 5-sums").

However, in general the average degree in a $$K_t$$-minor free graph can be large - Kostochka, Thomason, ... - so the "exceptions" will get really bad for large $$t$$.

I'm interested in this from a slightly different perspective. Are explicit descriptions known (or potentially tractable) of $$K_t$$-minor free graphs for general $$t$$ provided that $$n$$ is quite small relative to $$t$$?

So a desired result would be something like:

For any natural number $$t$$ and any $$t-2\leq n \leq 2t$$, every $$K_t$$-minor free graph $$G$$ on $$n$$ vertices has $$|E(G)|\leq (t-2)n-\frac{(t-1)(t-2)}{2},$$ unless $$G$$ is ... some exceptions like those in Jorgensen/Song and Thomas ...

If this is somehow easy then replace the vertex bound of $$2t$$ with something bigger.