Whilst it's true that CIGA makes it much more difficult for suppliers to terminate when a customer becomes subject to an insolvency procedure, there are still some cards that suppliers can play, depending how the situation develops.
As noted above, CIGA prevents a supplier making further supply conditional on payment of any outstanding amounts and effectively wipes the slate clean during the insolvency procedure in respect of pre-insolvency events (of any type) giving rise to termination rights. However, if a breach which would have entitled the supplier to terminate prior to the customer's insolvency recurs after the insolvency process begins, it would be regarded as a fresh breach and the supplier could (depending on the terms of the contract) terminate on that basis.
Suppliers which are prevented from terminating on insolvency must also continue to be paid for the supply provided during the insolvency procedure, failing which they will be able to terminate according to the terms of the agreement. So even in a situation where pre-insolvency, some invoices have gone unpaid, the supplier may find that once an insolvency procedure has commenced (and assuming the insolvency office holder wishes to continue the contract), the funds will be found to pay it going forward in order to avoid termination – albeit that any unpaid pre-insolvency invoices may have to wait until the customer emerges from the procedure.
In some cases, suppliers may also find that the insolvency practitioner or the customer (depending on the type of insolvency procedure) does not wish to continue the contract and is therefore content to allow it to be terminated (indeed, CIGA expressly provides for this). Lastly, CIGA allows the supplier to apply to court for permission to terminate on the basis that the continuation of the contract would cause it "undue hardship".