Say I have a simple random sample (without replacement) of 50 items from 500. Now I decided I want to increase my sample size to 60 items. I know that I can use my original seed number to generate a sample of 60 where the first 50 are my original sample.

But what if I don't have my original seed number. Can I generate a sample of 10 from the 450 that weren't already selected and combine the two samples so that overall I have a random sample of 60 from 500?

It seems to me this should work- there is no chance that an item will be selected twice and the probability of selection for the 51st item selected is 1/450 in either scenario. But I've never seen it done this way and I always see it emphasized that you should record your seed number for this purpose.

I'd greatly appreciate any comments/corrections on my thought process. Thank you!