For Congress that’s always how special elections end up. Otherwise over time the entire House would have a fragmented election schedule, with varying two year terms from district to district.
In the Senate, it was very explicitly by design that roughly one third of the Senate be up for election every two years (they had to stagger the terms of the first Senate, with some serving 2 years, some 4, and some 6) so special elections to fill Senate voids can only award the remainder of the term to keep things properly synchronized.
While I love the Founders, they made the relatively boneheaded decision of making Presidential succession ill defined. It says that the Vice President assumes the duties of the Presidency upon incapacity or death of the President (although how incapacity could be ascertained or determined is not specified whatsoever); but it also provides for a special election that could be held if both offices are vacant. Even worse, under some interpretations of the original text of Article II, it could be inferred that the Vice President himself could be viewed only as an Acting President and that Congress would have to craft law to establish a special election to fill the Presidential vacancy if a President died in office.
If we had gone down that route, then whoever would have been elected would have been elected to a full four year term, since nothing in the Constitution would have allowed for electing someone to the Presidency for a partial term. This would have permanently changed the synchronization of our elections and could have resulted in Presidents being elected in odd numbered years or other such foolishness.
As it actually played out, the first man to succeed to the Presidency, John Tyler, basically insisted he be sworn in as real President and argued that it was his right under the Constitution. My understanding is he basically walked into the White House, swore the oath, and took over. Not everyone agreed with it and it wasn’t entirely settled constitutionally, but no one really contested it so the precedent was established. Tyler’s motivations primarily were the fact that he knew his own party wouldn’t have even nominated him in a special election, so this was his only way to get into the White House. As it turned out, his extreme unpopularity with both Democrats and Whigs resulted in him basically getting very little done as President. He was called “his Accidency” by his political opponents (who basically was everyone on both sides of the aisle) and was formally expelled from his own political party while in office.
In the next Presidential election he established a third party to try and keep the White House, and possibly moreso to hurt the Whig party’s chances of winning (as he was probably still fairly unhappy about being expelled from the party.) It ended up that Tyler instead of running a campaign just spent a lot of time with his new wife, and he found that he had no real public support as a candidate. Finally he withdrew from the campaign when the Democrats informed him that his candidacy probably hurt them more than it hurt the Whigs, as Tyler’s views throughout his life were much more in line with the Democratic party than the Whigs (which is a big part of the reason he never got along with his own party) any split he caused would probably be in Democratic voters.
Later in life Tyler injured his already horrid reputation by openly embracing Virginia’s secession and becoming a member of the Confederate Congress. Although he died early in the war before the Confederate Congress’s opening session.
