Since 3.7 grains was mentioned as the starting load and it came from Hodgdon, I am guessing you have a 125 grain lead bullet with a flat tip on a round nose profile, like the one that went with the data.
Other bullet shapes or compositions will need different loads. 38 Special is rated for 17,000 psi. From Hodgdon's data, the pressure from your 3.7 grain load of powder gives 14,300 psi, and from 4.2 grains gives 17,000 psi (the maximum). By using the ratio of the logarithm of the ratio of those two pressures to the logarithm of the ratio of those two powder charges, we can calculate a charge of 5.5 grains will have about 24,600 psi, which is well over 38 Special maximum. Fortunately for you, it was not over the 35,000 psi maximum for 357 magnum.
The velocities of those three charges, 3.7 grains, 4.2 grains, and 5.5 grains, will be 982 ft/s, 1,076 ft/s, and 1,307 fps from the 7.7" test barrel used by Hodgdon (I don't have your barrel length to estimate your velocity from). By comparison, the powder called CFE Pistol would reach the same 1,307 ft/s with a heavier 6.9 grain charge, but the peak pressure would only be 17,800 psi, which exceeds the 38 Special limit but not the 38 Special +P limit, meaning it would be safe in a 38 Special, though it would tend to beat up a lightweight 38 Special over time.
The CFE Pistol load gets the same velocity and in the same ballpark of recoil and feel with much lower pressure (17,800 psi), while loading 700X to that same feel gives you a much higher pressure (24,600 psi). So feel doesn't tell you what's safe.
Normal procedure is to work a starting load up in 2% steps while watching for
pressure signs. This is because pressure grows exponentially as you increase powder charge. In other words, if you double powder charge the pressure more than doubles. How much more varies with the powder and its characteristic burn curves.