Blender has settings to help with this.
Let's start with 250 frames at 24fps
Let's say we want to get 4 times as many frames for the same time. Change the end frame to 1000, the frame rate to 96 (you could also leave the framerate and alter the framerate base to 0.25) and everything plays faster. Just under the framerate we have time remapping, change the new value to 400 as we have done 4x to everything else. Now when you render you will get 1000 frames and the animation will cover the same amount of movement.
Warning- You will notice that this feels "not fully supported" throughout blender, a keyframe in the timeline will remain at frame 250 while the timeline is highlighted to show the render ending at 1000, the current frame marker will only move to 250, as you try and position the current frame in the timeline, dopesheet or graph editor the frame marker will lag behind the cursor. It can be strange to have the frame move to nowhere near the cursor until you remember why. I recommend you alter these settings only during rendering.