There's a couple of solutions and I'll touch on them briefly. They can also be combined.
Option 1: Add a flat value
This produces the entire bell curve shifting left (or right! if supplied a negative min value). While it will exclude items at one and of the range, or the other, this may be beneficial (imagine a scenario where the high/good stuff needs to be common at some point and the maximum increased: during high level play you actually want to exclude the lower value items entirely!)
Option 2: Roll Twice, Pick Better
D&D 5th uses this type of mechanic with regards to advantage/disadvantage: roll twice and pick the higher (advantage) or lower (disadvantage). It has an on-average effect of being the same as adding +5 to a d20 (so about 20% better) but still generates values across the whole continuum (the bell curve gets squished to one side).
Option 3: Inverse Gausian
While this may not be the best solution for you there is in fact an Inverse Gaussian Distribution (or "Wald Distribution") which happens to look like an extremely squished bell curve (smashed up against the lower bound of 0) but the exact curvature can be tweaked to one's desires (but the maximal probability will occur towards the 0-end unless you do a
1-random). I use this one currently in a game I am working on so that things are very consistent with extremely rare deviations, but those deviations, when they occur, are very wild1. Code for how to compute this distribution can be located, though you'll have to piece it together from various sources (it relies on the "error function" which you have to look up independently).
1This means that the player will largely be able to rely on their skills, provided that they are evenly matched +/-1 with the enemies, but still face occasional failures, while as the disparity grows, things become more and more tenuous very rapidly. At +0 the player has a 98% chance of success. At -1, it drops to 92%. At -2, 80%, -3 50%. If the player plays smart they can still survive at a -3 disadvantage (conversely, at a +3 advantage their success rate only rises to only 99.5% or so! So still not a guaranteed success, although it will be very easy).