Comment by DonsDiscountGas

Most jobs I've had, I got by cold applying to a website. In a few cases recruiters reached out to me by email/LinkedIn (this is also fairly unreliable). The approach you're describing is definitely higher yield and you absolutely should cultivate and use a network, but all is not lost without a referral.

In answer to the overall question, I think most places leave a job posting up for a few days/week at least just as standard practice. Within that window I don't think it matters.