Playing a simple system, he has only two: pass 3NT if he has only 2-card support for responder's spades; return to 4♠ if he has 3-card support. Responder has, apparently, denied any interest in going any farther than game by making a nonforcing 3NT call.
All true. But really, responder has only shown willingness to stop in 3NT if opener has no spade fit; he hasn't denied interest in 6♠ if a spade fit exists. Recall that 2NT-3♥-3♠-4♠ is usually defined as a slam try by responder, because of responder's failure to use Texas instead of Jacoby.
Opener doesn't know how strong responder is. Most of the time, responder isn't going anywhere. But those 4♣, 4♦, and 4♥ rebids are lying idle. ("Maybe 4♣ shows 2-2-3-6 shape," I hear you say. Really? You would run from 3NT, with no guarantee of being able to make 4NT or 6♣, just because you had an offshape 2NT opener? I don't think you should.) If those rebids mean anything at all, they should show a "3-card superaccept of spades."
What does a "3-card superaccept" look like? In Part I we talked about what kind of a holding was necessary for an immediate superaccept after 2NT-3♥: it required a pure hand, with at least 4 trumps and at least 3 aces. The purity requirements were because we had to be afraid of being set in 4♠ opposite ♠xxxxx ♥xxx ♦xxx ♣xx. Now that partner has voluntarily bid game, that's no longer our biggest concern; we now just want to alert partner to the chance that a 30- or even 28-HCP slam may exist.
Over 1NT, the answer is generally no: if we don't have 9 trumps, and nobody is pushing us, why would we voluntarily choose to risk being set at the 3-level.
Over 2NT, in principle the answer could be yes, if we identified a pool of 2NT openers that had good game prospects opposite a hand with 5 spades and 0-3 HCP. Do such hands exist? Yes, but they are extremely rare. Simulations showed about 3% of 2NT openers with 3-card support, vs. about 35% of 2NT openers with 4-card support, had better than a 50-50 shot at game opposite a partner who would pass 3♠ given the chance.
Those handful of hands worth a superaccept were generally 21-HCP 5-3-3-2 hands, with very solid 5-card side suits and no wastage in the 2- and 3-card suits. The five hands I found (out of a thousand candidates) with a 60% chance at game opposite a weak partner:
Here are 5 more hands with about a 53% chance of making game opposite a weak partner:
These are similar to the first five, but have slightly less solid 5-card suits, or single finessing positions (A-Q-x or K-x) in a side suit.
If you wish to superaccept using the methods of Part I with these hands, be my guest. But I don't claim to have good enough judgment to superaccept on the above hands, but go low on ♠A54 ♥AK6 ♦AK762 ♣K9 which pencilled out to only a 44% chance of success.
Instead, I suggest than on hands like this — 5-loser hands with 5-3-3-2 and HHxxx (or perhaps AJTxx) in the 5-card side suit — you start with a simple completion of the transfer, but over 2NT-3♥-3♠-3NT, show your HHxxx suit instead of just correcting to 4♠ at your third turn. This alerts responder to the possibility of a slam if he thinks he can run both 5-card suits.
If you read Part I, you might be wondering, "why doesn't he recommend 4♣ on the third round on almost every '3-card superaccept' to leave responder space to explore?" Partly because responder has already passed up a chance to do that exploring: if responder has a two-suited hand, it can go 2NT-3♦-3♥-3♠ or 2NT-3♥-3♠-4♣ (more about that in Part III.) If responder has a flat 11-count it can go 2NT-3♥-3♠-4NT (quantitative). The hardest problem for responder is when he has a 5-4-3-1 hand and doesn't know if opener has wastage opposite the singleton. (We can't fix that entirely; even if we always bid 4♣ when we had nice 3-card support, there aren't three bids below 4M available for responder to use as splinters.) Able to solve only one problem, showing the potential second source of tricks felt better than having opener show the cheapest suit in which he'd be happy to hear a splinter from his partner, and better than simply cuebidding.
After 2NT-3♥-3♠-3NT, opener's 4♣, 4♦, or 4♥ shows a strong 5-card side suit, 3-card support for spades, and no wastage in th other two side suits. After opener's bid, responder can sign off at 4♠, RKC with 4NT, or cuebid (likely looking for 2nd round controls.)
After 2NT-3♥-3♠-3NT-4♠, responder's 4NT is RKC, 5 of a new suit is a splinter (rare, but not as rare as Exclusion or a control-ask.)
After 2NT-3♦-3♥, it is easiest to accept the loss of one step, and still have responder's 3NT be natural. If opener has a 5-card minor, he proceeds as above. With 5 spades, too bad; opener never drives past 4♥ on his own unless responder has made a slam invitation.