Vihje oli vähällä jäädä pöydän alle.

This is the fixed expression olla vähällä + infinitive.

It means to be close to doing something, to almost do something, or to be on the verge of doing something.

So:

• on vähällä pudota = is about to fall / almost falls
• oli vähällä unohtaa = almost forgot
• oli vähällä jäädä pöydän alle = was close to ending up / being left under the table

In many contexts, English will simply use almost.

Vähällä is a fixed part of the idiom olla vähällä + infinitive. It is in the adessive form, but for learners the most important point is that this whole expression should usually be learned as a chunk.

So it is best to remember:

• olla vähällä tehdä jotain

rather than trying to build it word by word.

You will see the form stay the same while olla changes for tense and person:

• olen vähällä myöhästyä
• olimme vähällä eksyä
• hän oli vähällä kaatua

The past tense is shown by oli, which is the past tense of olla.

In Finnish, when you have a structure like this, the finite verb carries the tense:

• on vähällä jäädä = is about to / is close to being left
• oli vähällä jäädä = was about to / was close to being left

The infinitive jäädä stays in its basic form; it does not change for tense here.

Jäädä often means things like:

• to remain
• to stay
• to be left
• to get left behind
• sometimes to end up

In this sentence, it is not just simple stay in the English sense. It is closer to be left or end up.

That is very common in Finnish. For example:

• Kirja jäi kotiin. = The book got left at home.
• Avaimet jäivät autoon. = The keys were left in the car.

So here jäädä pöydän alle suggests that the hint nearly ended up there or nearly got left there.

This is about the difference between result/movement and location.

• alla = under in a static sense, already there
• alle = to under / underneath, or ending up under

So:

• pöydän alla = under the table
• pöydän alle = to under the table / to a position under the table

Even though jäädä is not a movement verb like go or put, Finnish often uses alle when something ends up under something as the result of an event.

So jäädä pöydän alle means roughly end up under the table or get left under the table.

If you said jäädä pöydän alla, that would sound more like remain under the table, with more emphasis on an already existing location.

Because alle is a postposition, and postpositions in Finnish often require the noun before them to be in the genitive.

So:

• pöytä = table
• pöydän = of the table / table's (genitive form)
• pöydän alle = under the table / to under the table

This is very common with postpositions:

• talon takana = behind the house
• oven edessä = in front of the door
• tuolin alla = under the chair
• pöydän alle = under the table

Because the expression olla vähällä takes the first infinitive: the dictionary form of the verb.

So the pattern is:

• olla vähällä + tehdä
• olla vähällä + pudota
• olla vähällä + nukahtaa
• olla vähällä + jäädä

That is why jäädä appears in its basic infinitive form here.

Yes, the meanings are very close, and in many situations both are possible.

For example:

• Vihje oli vähällä jäädä pöydän alle.
• Vihje melkein jäi pöydän alle.

Both mean that the hint nearly got left under the table.

But there is a slight nuance:

• melkein is the general, neutral word for almost
• olla vähällä + infinitive can sound a bit more like a near miss or came very close

So oli vähällä jäädä may feel slightly more dramatic or event-like than melkein jäi.

Very often, yes.

This construction is commonly used for things that were not supposed to happen or would have had a negative result:

• Hän oli vähällä kaatua. = He almost fell.
• Olin vähällä unohtaa avaimet. = I almost forgot the keys.
• He olivat vähällä myöhästyä. = They almost were late.

So in this sentence, it naturally suggests that the hint being left under the table would have been accidental or undesirable.

It is not an absolute rule, but that is a very common feeling of the expression.