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This paper of Hasse deals with finding prime $q$ such that $ord_qp$ is even for some (fixed) prime $p$. Can some one suggest an online copy of it ? Or any survey in which its main ideas are highlighted.

H. Hasse, Über die Dichte der Primzahlen p, für die eine vorgegebene ganzrationale Zahl $a \neq 0$ von gerader bzw.ungerader Ordnung mod. p ist, Math. Ann. 166 (1966), 19–23.

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Among the first things which are quite natural to try, is to search in Google. For scientific papers Google Scholar might be sometimes more suitable. Notice that in Google Scholar you can often see link called all versions - if there are several versions available on line. In this specific case you get this.

Either way, you find this eudml link from which you can get to freely available version of the paper at GDZ. (You can either read the paper online or download pdf.

You can find several posts containing some advice on how and where to search for papers, for example, answers to this post: Finding a paper.

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