Unless you are familiar with the song that the soloist is singing, there really is no theory to it. The only thing u can do is use your ear and try to anticipate where the soloist is going. I ain't going to lie, soloing is a hard job. It is really scary too cuz for one thing it's just the singer and you on the piano, that's it. All eyes and attention is on yall. You got hundreds of eyes staring at you, watching and listening to your every move. If one of yall mess up, everybody gonna know it.
I'll say this, if you ain't mentally prepared for it, don't get up there. It ain't nothing worse than a great singer with a poor musician playing with them, or vice versa.
My advice is to study and practice progressions in every key. When you get up to play behing a soloist and you are unfamiliar with the song, take a minute to listen to what she's singing first. As you are listening, see if you recognize some familiar progressions. If you have no idea what to play, don't play anything. If you sorta got an idea, don't play anything, keep listening and maybe it will come together for you. If you know what to play, then jump right in and go for it.
I had an experience this past Sunday when a man got up to sing a solo. He started singing "I Need You Now" by Smokie Norful, but he didn't have no music with him. At first he was singing acappella, then people started asking me to go play behind him. Now, I have practiced the song before so I know the chords, but I was scared to death to go play. Anyways, I went up and played and it turned out great. It is a scary experience, but the more you do it the less scary it will be.
I don't know if this helps or not, but it's kinda hard to explain. Soloing can't really be taught, only learned.
