I previously overstayed my tourist visa on my last visit: can I still get a temporary working visa now?

A person in the U.S. who is in valid nonimmigrant status may apply for a change to another nonimmigrant classification. The key here is that at the time the application is filed, the person’s period of authorized stay must not yet have expired. The period of authorized stay is determined by the date to which a person had been admitted. This is generally found in the Form I-94, Arrival/Departure Record, which is issued to all nonimmigrants when he or she is allowed to enter the U.S. after inspection by an customs or immigration official at the port-of-entry. It is a small, white card that is stapled to the passport. For example, a tourist is normally given an I-94 valid for 6 months from his arrival. If within that 6-month period, the tourist is offered a job as a temporary worker and he or she is qualified, an application to change status from tourist to temporary worker based on the job offer of the prospective employer may be filed. If everything else is in order, the requested change of status is approved. A new visa, however, will have to be obtained from a U.S. embassy abroad to re-enter the U.S. if he or she leaves after the change of status is approved
But if the application is filed after the period of authorized stay as indicated in the I-94 has expired, then the requested change will be denied because the applicant is no longer in status.
The same is also true for applications to extend the period of authorized stay.
Staying in the U.S. beyond the expiration of the authorized period of stay is not the only cause for a person to not be in valid status. Working without government permission for example, is also a  violation of status. Any application for a change of status or extension of stay by a person who worked without employment authorization will also be denied because he or she is not in valid nonimmigrant status – even though the period of authorization has not yet expired!
What happens when the application for change of status is denied? This is your question.
From what you tell me, your prospective employer’s petition for you as a temporary worker was approved, but the application to change your status from a tourist was denied. A lot will then depend on how long you were out of status. Under U.S. immigration law, a person who has been out of status for more than 180 days is barred or disqualified from getting any kind of visa for 3 years. If one has been out of status for more than a year, the bar is 10 years. Also, any visa issued to such a person is automatically void whenever the person overstays even if for just for 1 day. This means that even a 10-year multiple entry visa is no longer any good in the case of any overstay.
It must be emphasized that the voiding of the visa is automatic – by operation of law. Thus, there is no need for any mark on the visa itself that it has been cancelled. A person who still tries to use such a visa is committing immigration fraud or misrepresentation. That is a kiss of death under U.S. immigration law.
Going back to your question, if the overstay is less than 180 days, you are still eligible to get a visa. So, if you are otherwise qualified, you will be given one. As a practical matter, the closer your overstay is to the 180 day mark, the harder it will be. A person with only a few days overstay is in a better position than one who overstayed a few months. Although neither will be disqualified if both overstays were for less than 180 days, the U.S. consular official who will interview the latter will probably need a lot more convincing.
Hope this helps.