Whether you can get mortgage help depends on whether you are entitled to a means tested benefit - in your case, income based job seekers allowance.
You may be entitled to this while you are still getting contribution based JSA if the total savings of you & your partner (if any) are less than �16,000 & if the calculations related to your joint incomes & the allowances made (including your mortgage costs) are low enough. If this is the case, then you should start getting mortgage help 13 weeks after your CBJSA started.
If you cannot get any meaningful answer from, JC+ then go to your local CAB, who should be able to do calculations to let you know whether you do have an entitlement.
If your savings are above �16,000 you will not be entitled to IBJSA or mortgage help, even when your 6 months of CBJSA runs out.