I like this. Could you expand a bit, please?
The joint resolution that annexed Texas and admitted it as a state included a provision that Texas could divide itself into up to five states:
New States of convenient size not exceeding four in number, in addition to said State of Texas and having sufficient population, may, hereafter by the consent of said State, be formed out of the territory thereof, which shall be entitled to admission under the provisions of the Federal Constitution
Texas had a fairly low population in the 1850s and 1860s, but enough to make multiple states (212,000 in 1850, and 604,000 in 1860, with about 30,000 being the usual minimum population for statehood (enough to form a full Congressional district)).
One scenario in which they might have exercised it would be the debate leading up to the Compromise of 1850. Up until the admittance of California as a free state, there had been an equal number of slave states and free states, giving slave states parity with free states in the Senate. When California petitioned for admittance, there weren't any good candidates for admission as slave states, so Northern Senators wound up giving a number of concessions in the Compromise of 1850 in exchange for the South acquiessing to the loss of Senate parity.
An alternative outcome might have been for parity to be maintained by dividing Texas into two states, perhaps explicitly as part of the Compromise, or perhaps in retaliation for passing the California enabling act with little or no Southern support. The issue of division of Texas did come up in the OTL compromise negotiations, but more in terms of splitting off unsettle sections of Texas to expand New Mexico Territory rather than forming new states (in exchange for Federal assumption of Texas debt).
If Texas did divide itself to maintain parity with California, it would likely then feel the need to divide itself again to match Minnesota (admitted OTL in 1858), and again to match Oregon (admitted OTL in 1859).
At some point, Northerners will get tired of this and attempt to return the favor, particularly if parity in the Senate were to skew slavery policy in a more pro-slavery direction. Pennsylvania, New York, and Ohio each had more than double the population of the largest Southern state, making it hard to argue against partition if the state legislature concurred. Judging from the Senators each state appointed in 1856, Republicans had a majority in the legislatures of all three state by then IOTL, making partition petitions at least vaguely plausible. They'd probably be vetoed or voted down in the Senate (with the help of the VP's tiebreaking vote), but once Republicans captured the White House, it'd be hard to stop the partitions from going through.
Once a norm develops of dividing existing states to obtain additional representation in the Senate, the political impetus to continue such divisions is going to be fairly strong. Taken to its logical extreme, there'd be one state for every one or two congressional districts, since the average population of a congressional district is traditionally the threshold population for admitting a territory as a state.