No, you cannot contribute to an account designated as an HSA once you no longer have a HDHP. You can continue to maintain the account until the funds are used up, but you cannot add funds to the plan when you are not eligible.
An account could be set up for the purpose you indicate, in the same manner, but it must be a regular bank account and not an HSA.
For clarification the IRS definition of a HDHP is shown below.
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