Rolling/Rerolling Jack

Honestly, it’s all just luck at this point. With the Jeanne d’Arc (Alter) pick-up banner coming really soon, you might be able to pick Jack up during that time. In the meantime, have you considered summoning/using another servant to fill for her until that one fateful summon? Hassan seems like a fair choice given his utility of generating a good amount of C-stars (though not as much as Jack, but still respectable given his 2 star rating and easy availability).

Other than that, be sure to save up your Quartz to summon her! (I know I am for Jalter :-) )

She got her rate up on Valentine last month, right? Every non-limited servants usually get rate-ups on annual events, when a new singularity comes out (the infamous singularity banner), in some events as fillers, or in a class focused banner, in Jack's case, Assassin's banner.
You can just wait it out. Besides, afaik, Jack isn't a story locked servant, so, she is pretty much available in every banners you summon at, cmiiw though.

She will get a rate up... I don't know why you say she wont.
There should be a class pickup summon in September, where on the Assassin(+Extra) day Jack and Jeanne d'Arc are the only two 5 stars you can summon.
Afterwards in December there should be a Christmas event banner where she will get a rate up.
The best chance you will have to get her will be the Assassin only pickup summons, but there are quite a few other banners where she will have a rate up.

She probably will have a banner for christmas rerun event