I'm assuming you mean that you metered the gray card and opened up 1-stop instead of 1/2-stop, so 1/2-stop extra exposure.
Exposure changes placement on the film curve. You will have more shadow detail on the film and each subject in the scene just slides up the film curve 1/2 a stop.
No problem, no adjustment to development, your Tri-X has plenty of room (latitude) on the film curve for that error and more.
All that that means is that when printing it will take a little bit more paper exposure in the enlarger to get the same print, nothing more.
What actually happens for most photos we mortals take is that we rarely use all the info on the film in a straight print. That extra 1/2 stop of shadow detail on the film (in fact probably more than that) is simply printed as black on the paper.
That's a choice not a given, remember that we gave the paper a bit more exposure, that extra exposure can be withheld in general but that changes the rest of the print, placing other subjects too light on the print, or you can dodge just the shadow areas.
Your original assumption that extra development was called for because of that low contrast scene is good, only the degree is in question.
Changing development, flattening the curve by using less development time would allow that extra 1/2-stop of detail to print. But, it also makes your print less contrasty, less snap. In your overcast situation it just makes your print look even flatter, lifeless. Probably not what you want.
The take away here is that the film curve needs to match what's important in the scene and the paper in the enlarger, not necessarily the exposure you use.