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]

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.

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.

In such a scenario I would say "That's a long shot." when I heard their theory for the first time.

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.

The phrase "Take a button and sew a vest on it" expresses this nicely.

(Erle Stanley Gardner was fond of this phrase. I can't recall seeing it elsewhere.)

Conjecture

NOUN
An opinion or conclusion formed on the basis of incomplete information.

VERB
Form an opinion or supposition about (something) on the basis of incomplete information.

Source: Oxford English Dictionary (Lexico link)

Not exactly an idiom, but a familiar saying in some circles is "multiplying causes beyond necessity", which in the negative injunction- "Do not multiply causes beyond necessity", is the guiding principle called Occam's Razor. This may not be useful for your purposes, but it is precisely about inferring more than is warranted by the evidence.

An idiomatic and alliterative expression might be:

An inflated inference.

to presuppose

to accept that something is true before it has been proved