For a while during the dynasty years it was fun to suss out NYY bandwagon jumpers by asking them who was the SS immediately prior to Jeter.
Real fans knew it was Tony Fernandez, but an amazing (or not) number of self-proclaimed "long-time" Yankee fans were totally stumped by that one even just five or six years into the Jeter era. Not sure some of them knew there even WAS a short-stop who pre-dated Saint Jeter.
Prior to that, one looking to find an answer to your 'two in a row' query has to wade back through the seasons anchored by Mike Gallego (1994), Spike Owen ('93), and Andy Stankiewicz ('92) (and, yeah, I had to look all that up). I won't say who the main guy was for more than a single season since I'm cheating here - but the answer goes back to the '89 - '91 seasons
Yep, the answer is Alvaro Espinoza, who had never played more than 37 MLB games in a season before becoming the regular NYY SS in 1989 at the age of 27. He played 146+ plus games at SS each year from 89-91. While his .599 OPS during the time pales in comparison to St. Jeter's batting stats, he did rank 4th, 1st, and 4th among AL shortstops in range factor per 9 innings those 3 years, so he helped out with the glove.
Espinoza was originally signed by the Astros way back in 1978. They released him after 2 seasons in the GCL where he failed to bat higher than .219. The Twins signed him and he worked his way up their system, getting a handful of ABs in the majors from 84-86, before being released after spending all of '87 in AAA. He made just 3 plate appearances for the Yanks in '88 and hit a modest .246/.262/.306 for AAA Columbus, but nonetheless won the starting SS job the next spring, succeeding Rafael Santana.
The Yankees released Espinoza in March of '92, after he hit .256/.282/.344 and finished 4th among AL SS in Putouts, 3rd in Assists, and 2nd in Double Plays in 1991. He spent all of '92 on the Indians' AAA club, batting .300 with 36 2B, 6 3B, and 9 HR en route to a very nice .455 SLG% and 79 RBI. He then played the next 4 seasons in the majors, mostly as a utility IF for the Indians, and went 1-for-2 in the 1995 World Series. In '96 he was traded to the Mets, along with Carlos Baerga, for future Astros DP combo Jeff Kent and Jose Vizcaino. He finished up his career in 1997 with the Mariners, ending with a .254 career AVG, 630 Hits and 22 HR over parts of 12 MLB seasons.
I hope everyone has enjoyed this little history lesson on Alvaro Espinoza that I for some weird reason felt like writing.