You should be aware that if you only have third party insurance you are not covered if your car is stolen or if you have an acciden that is your fault. Some car insurance companies do not offer third party only options.
Until your insurance company issues you with a cover note or a new policy then your car is not insured. That does not mean you need to have recieved your cover note. Many comapnies, give you instant cover, where the agreement is there imediatly.
Even if you find a cheaper quote than your existing insurance it's worthwhile telling your existing insurers to see if they will match or better the offer. You may find however that online quotes could be the cheapest.
