I would say that this isn't actually a case of one word with two readings, as you've suggested, but of two different words that are written in the same way. That distinction may seem trivial, but if you look up the two words in a good Japanese dictionary, it quickly becomes apparent that it is not.
大辞林 defines 入り{はいり}込む{こむ} as follows:
中にはいる。奥深くはいる。 「裏口から－・む」
入り{いり}込む{こむ}, on the other hand, is defined like this:
① 強引に中にはいって行く。はいりこむ。 「敵陣深く－・む」
② 物事が複雑に絡みあっている。入り組む。 「 － ・んだ事情」
③ 多くの人が寄り集まる。 「人の－・む事多ければ／仮名草子・浮世物語」
https://www.weblio.jp/content/入り込む
As you can see, while the first definition of 入り{いり}込む{こむ} is quite close to that of 入り{はいり}込む{こむ} (though not precisely identical), 入り{いり}込む{こむ} has two additional meanings that it does not share with 入り{はいり}込む{こむ} at all.
So when you encounter 入り込む without 振り{ふり}仮名{がな}, your task isn't so much to determine which of two readings to use, but to figure out which of two different words (which do have different readings) is intended. In most cases, context should probably allow you to do this.
