Selling a put means that you are obligated to buy 100 shares if you are assigned. That isn't hedging. It's increasing your long exposure.
To hedge, you would buy a put. If you did so, your long shares would be protected (not covered).
Put protected long stock is analogous to buying insurance for your house. The premium is the cost and the amount that the put is out-of-the-money is the deductible. If you want a zero deductible, you pay a higher premium. If you accept a $500 deductible (OTM strike price), the premium is less. It costs even less for a $1,000 deductible (deeper OTM put).
There are alternate ways to hedge. If you want to spend less money on protection and you are willing to accept less overall protection, you could buy a bearish vertical put spread. Like above, more protection (wider spread) costs more. This vertical would only be a partial hedge and would not help much if share price cratered.
If you owned 100 shares and you were willing to cap your upside gain, you could sell an OTM call and use the proceeds to buy an OTM put. This is called a long stock collar and typically done for little to no cost, depending on the strikes chosen.
In the end, it's a decision of risk and reward. Reducing risk costs dollars (buying a put) or it is obtained by opportunity loss (selling the call in the collar).