The volume of a tetrahedron is baseArea * height / 3. If we extend exactly two edges, they can only be adjacent. If you choose two that are not part of the base, the maximum height is obtained when the rotating edge is perpendicular to the base.
Then, the height coincides with the length of the edge.
The area of the base is sqrt(3)/4. Therefore, the maximum volume of the tetrahedron is:
Vmax = sqrt(3) / 4 * 1/3 = sqrt(3) / 12 = 0.144337567297406.
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