Binary Encoding of Ordinal Categories
The paper you've linked describes a method for predicting an ordinal dependent variable, but it sounds like you're looking for a way to use ordinal independent variables.  That part makes up just the first part of the paper's method, and is relatively easier.  It's somewhat analogous with one-hot encoding: you have indicators $x\leq \alpha$ for each level $\alpha$ (except the maximum one).

For example, if the levels are Low, Medium, High, your new variables will be "Low" and "Low or Medium" (skipping "Low or Medium or High" since that's always true).