Your rates are likely to go up again when your car insurance company finds out about your speeding ticket. As for your overall policy coverage, at renewal time you may find that you are receiving a non-renewal notice.
Your conviction for driving under the influence (DUI) was a year ago and thus your auto insurance provider has probably already looked at your driving record, to see it and rate you on it.
Any drug or alcohol-related violation is looked at as a major violation by your car insurance company, and so rates will go up, a lot, as you likely already found out.
If your insurance company somehow didn't already know about the DUI, then at renewal time (this time around) your rates will go up, if they decide to renew your policy.
It's not unusual for your premium amount to double or even triple after a DUI. Adding to that a speeding ticket, your rates will be quite high or these violations may be more than your car insurance company can take.
The combination of a DUI and speeding ticket may make you too much of a risk, according to your insurer's ratings system and underwriting rules. If this is the case, you may find yourself being non-renewed and have to search for new car insurance coverage.
If your rates go up immensely or your car insurance company drops you, all is not lost. Even with a DUI and other traffic violations on your driving record, you can still shop around and may even be able to find lower car insurance rates than what your current insurance provider is offering. Some insurers are willing to give first-time DUI drivers a break, figuring they'll be far more careful in the future.