DII has a waiting period of 6 months that's why CI can fill in the gap
No, Disability Income Insurance sold in Singapore has a waiting period (called the "pre-benefit" or "elimination" period) of anywhere from 2 months to 6 months, your choice depending on carrier, policy selection, and premium. (I recommend 90 days or longer, but 2 months is available.)
Critical Illness (CI) coverage doesn't fill any "gap" if it pays nothing, and it so often pays nothing because it only covers calamities that are specifically listed in the policy. If you get run over by a bus, for example, you could be very ill, and that could be quite critical, but CI won't pay even a penny, sorry to say. OK, then I suppose you could spend more money (and enrich your insurance agent) to buy "Personal Accident" coverage, and that policy has another enumerated list of calamities, which might include the bus example. But what if your calamity isn't listed there either? Ooops.
I don't think CI+PA can replace DII (or supplemented ElderShield for that matter). At best you could say they
might augment DII or supplemented ElderShield, but if you're concerned about a "gap," they won't reliably solve that problem.
So what can really fill a "gap"? Cash can! That's called an
emergency reserve fund, and it's important to have one. If you don't have one, try to save enough to create one, and stash most of it in Singapore Savings Bonds (SSBs). As a "rule of thumb," your emergency reserve fund should cover at least 6 months of normal household expenses if you were to lose all your income, no matter why or how. You can include your CPF Ordinary Account balance in this "rule of thumb" calculation, to the extent your OA balance can be tapped to service your mortgage (if you have one). Many people prefer to keep more than 6 months if they can -- this is just a "rule of thumb." And some people make this calculation based on "emergency mode needs," meaning they subtract out things like manicures, spa treatments, and European vacations from their normal household expenses. Regardless, figure out what your emergency reserve fund should be, and get it done if you can.
In many cases your emergency reserve fund might be partially or fully held at the "Bank of My Mother" or the "Bank of My Uncle." If that's how things work, reliably, and if everybody agrees to that arrangement, no problem. It's just one of the many advantages wealthy families have, if they wish.