Fiber jamming modules (FJMs) offer flexibility and quick stiffness variation, making them suitable for follow-the-leader (FTL) motions in continuum robots, which is ideal for minimally invasive surgery (MIS). However, their potential has not been fully exploited, particularly in designing and manufacturing small-sized FJMs with high stiffness variation. Although existing research has focused on factors like fiber materials and geometry to maximize stiffness variation, the results often do not apply to FJMs for MIS due to size constraints. Meanwhile, other factors such as fiber number and packing density, less significant to large FJMs but critical to small-sized FJMs, have received insufficient investigation regarding their impact on the stiffness variation for FTL deployment. In this paper, we design and fabricate FJMs with a diameter of 4mm. Through theoretical and experimental analysis, we find that fiber number and packing density significantly affect both absolute stiffness and stiffness variation. Our experiments confirm the feasibility of using FJMs in a medical FTL robot design. The optimal configuration is a 4mm FJM with 0.4mm fibers at a 56% packing density, achieving up to 3400% stiffness variation. A video demonstration of a prototype robot using the suggested parameters for achieving FTL motions can be found at https://youtu.be/7pI5U0z7kcE.