This question is in regards to stability, so that's what I'm going to focus on (as opposed to strength, which would require far more information regarding materials, dimensions and connections).
Your table has three configurations:
- Both wings down
- One wing (left or right) up, the other down
- Both wings up
Stability can be measured in three ways: how much horizontal or vertical (upwards or downwards) force is required to tip a structure over.

Without a doubt the most unstable of these configurations is with both wings down. In this case, the table is supported basically only by the four central feet. Obviously, very little horizontal force is required to tip it over on its long side. In fact, the force required is equal to
$$F_h = \dfrac{Ww}{2h}$$
where $W$ is the total weight of the table, $w$ is the width of the supports (in this case, 6"), and $h$ is the height of the table. Therefore, we can see that the stability is proportional to the table's weight and to how wide its supporting base is, and is inversely proportional to the table's height.
For downwards vertical loads, the equation is
$$F_{v-} = \dfrac{Ww}{2e^-}$$
where $e^-$ is the distance between the farthest point outside of the supports and the nearest support. Given that the feet of the table are basically right on the outer perimeter of the table, the value of $e$ is close to zero, therefore $F_{v-}$ is necessarily quite high.
For upwards vertical loads, the equation is almost the same
$$F_{v+} = \dfrac{Ww}{2e^+}$$
but $e^+$ is the distance between the farthest point outside of the supports and the opposing support. $F_{v+}$ is always lower than $F_{v-}$, but arguably also less likely.
These equations can be applied for the other configurations as well. The new legs dramatically increase the value of $w$ in the direction perpendicular to the 39" side, increasing $F_h$.
For downwards vertical loads, the design of those rotating legs matters. If they open as in the first drawing, then that's fine: once again the legs cover basically the entire perimeter of the table, leaving little space to flip the table and therefore requiring a massive load to do so. If it's a single leg swinging to the middle of the wing, then that's more problematic. In one direction (perpendicular to the 39" side), you'll have $e^- = 11''$ (half of 22"). In the other, you'll strictly speaking still be fine due to the main legs, but that will put a lot of strain on the hinges connecting the wings to the main segment (this is the instability you mention "in the corners"). Also, this second design requires at least two hinges per wing, preferably as far as possible from each other.
For upwards vertical loads, the same issues apply to the second design, but to a lesser extent. $e^+$ will equal $6+22=28''$ or $2\times 22+6=50''$ if one or both wings are open, respectively.