So we see that Rush was working hard on the 9th chevron project while his wife was dying. He never really spent enough time with her in the end. It seems to me like he doesn't want all that work he did while his wife was dying to be in vain. Like he has to follow through with the mission at all costs otherwise neglecting his wife all that time would have been for nothing.
In that case it could be he's not really after anything specific (particularly not ascension - at least not as his primary objective), but rather the discovery of the destiny and the knowledge within as a whole.
Maybe he's just hoping for any new and significant insight or discovery into the ancients regardless of what it is, in hopes that his work wasn't for nothing. (i.e. he can feel better for his actions at least)