I decompose the question into four parts, all of which Pete already knows a lot about, and/or are mentioned in the comments:
(1) The books by Malle-Matzat and Völklein thoroughly explain the workhorses of the field such as the rigidity method of John Thompson.
(2) Later results: Pete himself is responsible for a significant further result in the $A_1(2,p)$ case. It seems well-known that the case $A_1(2,q)$ (and $G(q)$ for other many other algebraic groups $G$) is difficult when $q$ is a prime power rather than a prime. (But that was the wisdom 10 years ago --- see the update below.) Pete already gave an interesting recent reference for the case $A_1(2,16)$.
(3) People tabulating results. It seems that the main activity is by Klüners with help from Malle. The on-line database of Klüners is organized by the permutation degree of the group, which is of course not the same as the cardinality. Also Klüners looks at all finite groups, not just finite simple ones.
(4) Sizes of finite simple groups. The Wikipedia page, list of finite simple groups, is excellent. It lists all 16 non-abelian finite simple groups of order less than 10,000.
So it could be best to correspond directly with Klüners concerning specific inverse Galois results.
If you're interested in organizing finite simple groups by cardinality --- this is a question that comes up from time to time --- it could be very useful to convert the Wikipedia page to a Python/SAGE program.
Update 2: David Madore had the same idea 8 years ago and wrote a code not in Python but in Scheme. So if you're interested in the smallest simple group not proven to be a Galois group, you only need to go down Madore's list. Note the Shih-Malle theorem that Pete summarizes in his paper: $A_1(2,p)$ is $\mathbb{Q}$-Galois when $p$ is prime and not a square mod 210. The first prime $p$ that is a square mod 210 is 311. (Pete's paper settles some other such $p$, but not that one.) Madore's table shows that the open case $A_1(2,311)$ shows up just a few entries after the only open sporadic case, $M_{23}$. So what does that leave that's smaller than $M_{23}$?
Update 1: Some Googling found an interesting paper by Dieulefait and Wiese and a more recent paper by Bosman. It seems that a lot more is known about the $A_1(2,q)$ case in recent years using the number of theory of curves Galois representations attached to modular forms, rather than the rigidity method of Thompson. In particular, no open cases remain in the Wikipedia list of finite simple groups of order less than 10,000, other than possibly $A_1(27)$ and ${}^2A_2(9)$, which is the unitary group $\text{PSU}(3,\mathbb{F}_9)$. Actually it looks like ${}^2A_2(9)$ doesn't survive either because (like $M_{11}$) it is handled by older methods, according to Malle and Matzat. The smallest finite simple group which is not known to be a Galois group over $\mathbb{Q}$ could be an interesting trivia question for the moment.
The short answer is (as far as I am aware) no, but there is a lot that is known. Jannsen and Wingberg have given an explicit presentation for $Gal(\overline{K}/K)$ in the case that the residue characteristic is not $2$ (published in Inventiones Math in 1982/1983), and Volker (1984, Crelle) handles the case when $K$ has residue characteristic $2$ and $\sqrt{-1} \in K$. This does not, however, make it trivial to determine which finite groups are quotients of $Gal(\overline{K}/K)$. Some more information can be obtained from Section VII.5 of "Cohomology of Number Fields" by Neukirch, Schmidt and Wingberg. Here's a paraphrase.
If $K$ is a local nonarchimedean field with residue field of characteristic $p$ (and order $q$), let $G = Gal(\overline{K}/K)$, $T$ be the inertia group, and $V$ be the ramification group. Then $G/T \cong \hat{\mathbb{Z}}$, $T/V \cong \prod_{\ell \ne p} \mathbb{Z}_{\ell}$, and $V$ is a free pro-$p$ group of countably infinite rank. Iwasawa showed that $G/V$ is a profinite group with two generators $\sigma$ and $\tau$ so that $\sigma \tau \sigma^{-1} = \tau^{q}$. Also, the maximal pro-$\ell$ quotient of $G$ is known for all $\ell$. For example, if $\mu_{\ell} \not\subseteq K$ and $\ell \ne p$, the maximal pro-$\ell$ quotient is $\mathbb{Z}_{\ell}$ (i.e. for each positive integer $k$, there is a unique Galois extension $L/K$ of degree $\ell^{k}$, namely the unramified one).
Best Answer
You should find what you want in the following article by Jochen Koenigsmann: The regular inverse Galois problem over non-large fields. J. Europ. Math. Soc., 6(4):425–434, 2004.