Yes. If there is a warrant showing from a local municipality (city jail) for tickets, the sheriff will hold you as well. It will be up to the city or the other county to come over and pick you up. At least in Tarrant County, most local municipal judges will give a defendant credit towards money owed for tickets (class C misdemeanors) for the time he sat in the county jail. But this is discretionary - not mandatory - for the city judges. In Tarrant County, the release desk will provide you with a print-out of days you've spent in TC Jail upon your release. Take that document to the municipal judge and ask to get credit for the days you just spent in the first county jail. If another county picks you up, request to have your sentence run concurrent with the plea deal (probation or jail sentence) you just concluded in the first county. Also be aware that visitors will also be checked for warrants when they come to see you in ajail. If they have any active warrants, they will be arrested.
The use of this information shall not create any attorney-client relationship. You should seek and retain professional counsel in your local county.
Typically yes. Of course jails do miss things, but if they see any warrants, standard procedure is for them to notify the agency that has the warrants to come get you. Warrants often pop up during bookout that weren't seen during bookin for some reason.
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The short answer is yes. The long answer is that you shouldn't ever bond out in one county if you can't make bail in ALL the counties where you have warrants. You don't want to spend the money to bond out of one county, only to not be released from custody AND not be getting ny credit for the time in custody with the county you bonded out of.