No, and kind of perhaps.
First, numerically, the two coding schemes are nearly the same thing. Calling something 1, 2, 3, 4, 5, and 1.5, 3.5, 5.5, 7.5, 9.5 is simply a rescaling, instead of a jump of 1, it became a jump of 2 between every level. In other words, if the ordinal variable is unfit as a continuous independent variable, it'd still be unfit even in disguise behind the "proper" unit. I said "nearly the same thing" because not all ordinal categories are of equal widths. In those cases, recoding sometimes may improve the situation (see below.)
Second, what determines if the said ordinal variable can be model as categorical or continuous is if it is linearly related to your dependent variable. So, keep it either in its orginal form or the recoded form, use scatter plot (or component plus residual plot if it's a multiple linear regression) with the aides of a regression line and a LOWESS curve to see if linearity can be safely assumed. If yes, then possibly fine to model it as a continuous variable. If not, then some more drastic recoding such as aggregation or even dichotomization may be considered.