The H&L calculator covers a different case, ie one where you contribute some post-tax money into a defined-contribution scheme, like a SIPP, and the manager of the scheme automatically adds basic rate tax relief. If you pay tax at 40% or 45%, you must claim additional tax relief separately (eg through your self-assessment).
What we are talking about here is different, it's making additional contributions to a defined-benefit scheme. The scheme has already calculated that to buy £1,000 of extra pension will cost you, say, £10,000.
If you pay in instalments through your salary, I'd like to think it should be done as salary sacrifice so there's nothing else to do.
If you pay a lump sum, out of your post-tax money, it's trickier.
If you earn £50k, no problem: contact HMRC and they'll refund the tax you paid on those £10k (probably after a self-assessment).
If you earn, say, £16,570, you will have paid tax on £4,000 (= £16,570 less the £12,570 after which you start to get taxed), and those taxes HMRC will refund. But you haven't paid taxes on £10,000, so how does HMRC refund you taxes you never paid in the first place?
Remember, this is not relief at source.
I'll admit I do not know the answer with absolute certainty, but I'd be interested in hearing from anyone who think they do - and who can elaborate.