TL;DR: No.

Faith, as it's commonly understood, means belief based on spiritual conviction rather than evidence or argument (or, to put it differently, belief based on confidence in that belief, which is somewhat circular, even if that confidence was bestowed by the thing you believe in).

Another definition of faith is just trust. I've commonly heard theists say that this is the definition of faith that they use, i.e. trusting God (whom they may believe in for other reasons). While it may sometimes be used in this way, it seems to more commonly be used to support a belief. Using it in that way would be using an element of a belief (God's traits) to support that belief itself (whether God exists) is much more directly circular. But that's somewhat beside the point.

Yet another definition of faith is a religion itself, but that's not really relevant here.

Hope can certainly factor into the first two definitions, because if you don't have hope about something, that lends itself to indifference, which is somewhat antithetical to conviction or trust. But the faith is not directly based on hope.

Beyond that, using "faith" for what you're describing would be creating a new definition of faith.

Belief that is itself (partially or fully) based on hope sounds like an appeal to consequences fallacy: I can certainly wish that my closet is a portal to Narnia, but that has no bearing on whether that's actually the case.

Although it can be rational to take precautions for some event or truth that you believe to be improbable, e.g. buying insurance in case an accident occurs. But probability and confidence and 2 different things, and I'd reject Pascal's Wager because it's using probabilistic reasoning for what I see as confidence (I'd say e.g. the probability of there being a unicorn in my backyard is zero, with some tiny non-zero confidence, rather than saying the probability is non-zero; the tiny non-zero confidence there doesn't factor into precautions I take at all, whereas a non-zero probability might factor into precautions I take). If you did land on a probability, maybe you end up with a different conclusion. Although this, in itself, can't convince you that it's true; it can merely get you to take precautions, as it were: e.g. someone may expose themselves to things that may convince them or they may suppress their doubts.

TL;DR: No.

Faith, as it's commonly understood, means belief based on spiritual conviction rather than evidence or argument (or, to put it differently, belief based on confidence in that belief, which is somewhat circular, even if that confidence was bestowed by the thing you believe in).

Another definition of faith is just trust. I've commonly heard theists say that this is the definition of faith that they use, i.e. trusting God (whom they may believe in for other reasons). While it may sometimes be used in this way, it seems to more commonly be used to support a belief. Using it in that way would be using an element of a belief (God's traits) to support that belief itself (whether God exists) is much more directly circular. But that's somewhat beside the point.

Yet another definition of faith is a religion itself, but that's not really relevant here.

Hope can certainly factor into the first two definitions, because if you don't have hope about something, that lends itself to indifference, which is somewhat antithetical to conviction or trust. But the faith is not directly based on hope.

Beyond that, using "faith" for what you're describing would be creating a new definition of faith.

Belief that is itself (partially or fully) based on hope sounds like an appeal to consequences fallacy: I can certainly wish that my closet is a portal to Narnia, but that has no bearing on whether that's actually the case.

We might consider this to be backwards: rather than belief based on hope, we might say that hope is based on belief. One might believe something to be possible, and from there, hope that it is true or would happen. This hope does not (directly) change our belief, we instead merely focus on a particular belief.

Also, it could be rational to take precautions for some event or truth that you believe to be improbable, e.g. buying insurance in case an accident occurs. But probability and confidence and 2 different things, and I'd reject Pascal's Wager because it's using probabilistic reasoning for what I see as confidence (I'd say e.g. the probability of there being a unicorn in my backyard is zero, with some tiny non-zero confidence, rather than saying the probability is non-zero; the tiny non-zero confidence there doesn't factor into precautions I take at all, whereas a non-zero probability might factor into precautions I take). If you did land on a probability, maybe you end up with a different conclusion. Although this, in itself, can't convince you that it's true; it can merely get you to take precautions, as it were: e.g. someone may expose themselves to things that may convince them or they may suppress their doubts.