Idiom meaning inferring too much from the available evidence
Solution 1:
Read too much into something
to think of (something, such as a comment or situation) as having a meaning or importance that does not seem likely or reasonable
Solution 2:
Jumping to conclusions
Jumping to conclusions ... is ... where one "judges or decides something without having all the facts; to reach unwarranted conclusions".
[Wikipedia]
Jump to conclusions
To make decisions or form opinions before one has all the pertinent facts.
- I know you found some suspicious things in her office, but don't jump to conclusions—talk to her first.
[Farlex Dictionary of Idioms]
Solution 3:
In a case like this I think I'd say:
You put (or added) two and two together, and got five.
https://dictionary.cambridge.org/dictionary/english/put-two-and-two-together-and-make-five
Although this expression does suggest that the inference is wrong, rather than only unjustified.
Solution 4:
Reading Tea Leaves
This is perhaps a bit more obscure than you're looking for, but it's a form of divination: you make some tea, drink it off, and then look at the pattern left behind by the leaves to infer the answer to some question you have. The expression is often used to describe the process of finding patterns in randomness and possibly using them to explain something unrelated.
Solution 5:
Both the aforementioned "reading too much into something" and "jumping to conclusions" are good idiomatic expressions for the meaning sought by the OP.
"Jumping to conclusions" can be expressed less idiomatically and more formally as "faulty generalization". The Wikipedia page cited currently includes some eleven different versions of this, all more or less formal, with the most idiomatic being "leaping to conclusions" and "secundum quid" (where the idiom in question is Latin).
There is also the rather idiomatic expression "making a mountain out of a molehill". This is more typically used to indicate when someone is exaggerating the importance or impact of a small problem, but it can be and is used on occasion to refer to what the OP asks about, drawing too many conclusions from too little evidence.