Sorry..I see that now, IL NG. Not sure why I was sidetracked to CA.
YES...I agree with you...IL DOES give pref points for promotion (to state workers). Here is a link to the statute http://www.ilga.gov/legislation/ilcs/fulltext.asp?DocName=006500050K10-1-16
And here is the specific language that applies
The civil service commission shall give preference for promotional appointment to persons as hereinabove designated whose names appear on existing promotional eligible registers or promotional eligible registers that may hereafter be created by adding to the final grade average which they received or will receive as the result of any promotional examination seven-tenths of one point for each 6 months or fraction thereof of active military or naval service not exceeding 30 months. The numerical result thus attained shall be applied by the civil service commission in determining the position of such persons on any eligible list which has been created or will be created as the result of any promotional examination held hereunder for purposes of preference in certification and appointment from such eligible list.
So it is tied to active service.
That is, the points are available, but not for guard service...only for active service. But if you were activated either Title 10 or Title 32, that would count
Now...on to your question...
Will the fact you had a prior general discharge prevent you from getting points?
Here is the language that applies:
Veteran's preference. Persons who were engaged in the active military or naval service of the United States for a period of at least one year and who were honorably discharged therefrom and all persons who were engaged in such military or naval service who are now or may hereafter be on inactive or reserve
duty in such military or naval service,
So your general discharge is not an issue. But you do need that active service.