I would make the offer but just spin it not as 2nd prize. Present it as a situation where 2 candidates really impressed the company, which would ideally hire both but can't. Say that although the company wishes it could bring this exciting candidate in permanently, at this time what can be offered is 2 months work.
For the unemployed person, this is an opportunity. Sure, it's also a bit of a disappointment as she didn't get the job she applied for, but some money, extra experience, perhaps some new references, etc. are good things. Also, it puts her in a better position with this place if something happens and there's an opening.
If it's framed right, it is not insulting to the applicant. Also, she might really need to get back to work, and perhaps she can't afford the luxury of hurt feelings and feeling insulted. It would be awful if, under the kind motivation of not insulting her, she were denied income she needs.
The employer could use her for the short-term. In the industry people do take short term positions; it isn't unheard of. Leave it up to her as to how delicate her feelings are. She might instead feel happy that she did well in the application process and is getting some work.