Yes, it would be possible that your rates would go up if you allowed your current auto insurance policy to get cancelled for non-payment and a month later decided to get insurance with a new company.

A new insurer may be wary to take your business since you have shown that you have previously not paid fully on your policy and allowed it to cancel for nonpayment. Being more of a risk in this way could cause an insurer to not either want to accept your business or place you in a higher rating tier.

Not paying your auto insurance could also have other consequences such as it being noted on your credit report that you were canceled for non-payment. And the month lapse in insurance would normally be reported to your state's department of motor vehicles. Most state's DMVs will then suspend your vehicle's registration for a time period and may require you to obtain a SR-22 to show future proof of financial responsibility.

If you no longer want auto insurance through your current insurance carrier then you should notify them and cancel your policy. However before doing so you will want to obtain a new car insurance policy to cover your car so that there will not be a lapse in coverage.