This is from Lay It On Me by Vance Joy "I found you hidden in plain sight, why'd it take so long" and I have always thought that one of the greatest romantic lines of all time is this one "There's something about you girl that makes me sweat" (Need You Tonight - INXS).

There were a few. "Rambling Rose" by Nat King Cole is one I can think of at the moment. 

"Since I Don't Have You" - Skyliners

"The Clouds" by the Spacemen
Booker T & the MG's had a multitude of instrumental hits as did Bill Black's Combo that were pretty funky
"Cast Your Fate To The Wind" first by  Martin Denny and a few years later by Sounds Orchestral
"Glad Rag Doll" by Johnny Maddox
"Midnight" by Johnny Gibson
Johnny & The Hurricanes is another group that had a number of great instrumentals.
 

Another rocker that was a hit twice was "Do You Love Me" by The Contours. It hit the Hot 100 Top 10, peaking at #3 in October 1962. It reappeared in 1988 when it peaked at #11 on the Hot 100 in August.

In 1960 the Rebels had a hit (really big in Canada) "Wild Weekend". It was re-released in 1963 and became a hit again, this time under the group name of The Rockin' Rebels. Originally it was used by Tom Shannon as his theme song for his show on WKBW Buffalo. I don't have this songs Billboard history.

In 1959 "Rockin Crickets" was a hit by the Hot Toddys which made the Hot 100 peaking at #59. The exact  identical song was re-released in 1963 this time by the Rockin' Rebels (same group-different name). It peaked at #87 on the Billboard Hot 100 and was the follow-up to the 1963 Wild Weekend.

In 1959 "Linda Lu" by Ray Sharpe made #46 on the Hot 100. It was re-released in 1963 and received airplay in Southern Ontario. Don't know if it made the Hot 100 the second time.

We all know "Donna" by Ritchie Valens from 1958. It was re-released in 1965, maybe only in Canada-I don't see it in the Hot 100, but it was a Top 10 hit here during the summer of 1965 

I don't remember the first 45 I bought. When I started buying records I bought 78's-something I regret now-until they were discontinued. The first record I bought was "One Step At A Time" by Brenda Lee. The first 45 might have been "Stairway Of Love / Just Married" by Marty Robbins, because I really wanted it and it was available and the 78 wasn't. I have been buying 45's up until recently as I collect them, and don't remember what the last one was.