Indeed, Oswald probably would have gone on to shoot the president. As it was, while evading arrest he had his chance encounter with Herman Wouk, who was about to give a talk at UT Dallas on his latest (and last) novel Youngblood Hawke. Oswald recognized the author, and with two Secret Service agents and three Dallas police officers thirty seconds behind him, drew a revolver and emptied it into Wouk.
After that came the sadness of 1964... What a gentlemanly election season Kennedy and Goldwater had planned, and oh how it turned into the polar opposite. After Goldwater's "soft on communism" accusations caused a sharp decline for the president in the polls, JFK adopted his infamous southern strategy: Southern Democrats accused Senator Goldwater of voting against the 1964 Civil Rights Act for all the wrong reasons, and pointed to his votes in favor of the 1957 and 1960 Civil Rights Acts. After expecting unusual southern support for a Republican, Goldwater found the South going solidly Democrat once again per the polls. And then came the revelations of Kennedy's infidelities, a decent into mudslinging no one wanted, and despite Kennedy leading the popular vote by a quarter of a million votes Goldwater narrowly squeaked out an electoral victory.
Now, Kennedy's second non-consecutive term of 2013-'17 is far too recent for detailed discussion here, but had Oswald succeeded fifty years earlier our first nonagenarian president would be the farthest thing from anything we'd consider plausible. Indeed, given JFK's earlier health issues we'd probably call it ASB territory to suggest he'd live into his 80s, let alone become the august centenarian we know in 2018.