If a hot Jupiter was not tidally locked, are there any specific cases where its wind speeds would be milder than those found on Jupiter?
After controlling for the age of the hot Jupiter, of course. The one thing with giant planets is that they are so warm on the inside that the temperature gradient between the interior and the surface is smaller than that on a cold gas giant like Jupiter (though maybe the temperature gradient between the interior and surface is so large that our surface temperature may not matter as much).
Now, what are the causes of wind? Wikipedia explains it below.
Wind is caused by differences in pressure. When a difference in pressure exists, the air is accelerated from higher to lower pressure. On a rotating planet, the air will be deflected by the Coriolis effect, except exactly on the equator. Globally, the two major driving factors of large-scale winds (the atmospheric circulation) are the differential heating between the equator and the poles (difference in absorption of solar energy leading to buoyancy forces) and the rotation of the planet. Outside the tropics and aloft from frictional effects of the surface, the large-scale winds tend to approach geostrophic balance. Near the Earth's surface, friction causes the wind to be slower than it would be otherwise. Surface friction also causes winds to blow more inward into low pressure areas.
Winds defined by an equilibrium of physical forces are used in the decomposition and analysis of wind profiles. They are useful for simplifying the atmospheric equations of motion and for making qualitative arguments about the horizontal and vertical distribution of winds. The geostrophic wind component is the result of the balance between Coriolis force and pressure gradient force. It flows parallel to isobars and approximates the flow above the atmospheric boundary layer in the midlatitudes. The thermal wind is the difference in the geostrophic wind between two levels in the atmosphere. It exists only in an atmosphere with horizontal temperature gradients.3 The ageostrophic wind component is the difference between actual and geostrophic wind, which is responsible for air "filling up" cyclones over time. The gradient wind is similar to the geostrophic wind but also includes centrifugal force (or centripetal acceleration).
Now - here's the thing: if a hot Jupiter was not tidally locked, then the initial temperature difference between the day side and the night side would not be so great (although it is true that the difference is not that big in tidally locked hot Jupiters - but then that low night-day temperature contrast is irrelevant to our question as it is caused by the wind).
One might initially believe that tidally locked planets do not rotate - but that is not true - they do rotate, and their rotation speeds are not negligible because their orbital periods are so slow. So theoretically, one could consider the case of a non-rotating non-tidally locked hot Jupier (which would only be possible in very young hot Jupiters due to the timescale of tidal locking at small distances - but those hot Jupiters are still possible).
Hot Jupiters that rotate at a very fast speed, though, would probably have a significant equator-pole temperature gradient that would drive a very fast wind speed.
So - back to my original question. If a hot Jupiter was not tidally locked, are there any specific cases where its wind speeds would be milder than those found on Jupiter?