What standard deviation is used for calculating standard error?

I take the following quotation from Wiki:

The standard error (SE) of a statistic (usually an estimate of a parameter) is the standard deviation of its sampling distribution or an estimate of that standard deviation. If the parameter or the statistic is the mean, it is called the standard error of the mean (SEM).

This means that the term standard error does not stand alone, it must be attached to a statistic and its sample distribution.

Also from Wiki for a population mean:

The sampling distribution of a population mean is generated by repeated sampling and recording of the means obtained. This forms a distribution of different [sample] means, and this distribution has its own mean and variance. Mathematically, the variance of the sampling distribution obtained is equal to the variance of the population divided by the sample size. This is because as the sample size increases, sample means cluster more closely around the population mean.

So, standard error by the definition is already involved repeatedly sampling.

Please note that the formula you quoted is for a population mean, see here for some examples.

Using sample standard deviation provides an estimate for the standard error of a sampling distribution. As you note, different samples will most likely have different sample standard deviations, leading to different standard error estimates. None of these are the standard error since none of them come from the true population standard deviation.

The bigger question is, what are you intending to do with these multiple samples? That will direct if there is more for you to do and what.

The best thing to do is to find an estimate of the standard deviation using all the samples. Based on how you draw those samples (such as SRSWOR, or Cluster sampling, etc.), you may find suitable estimates of $sigma^2$ that uses all the samples. Using all the samples will reduce the standard error for the estimate $hat{sigma}.$