With the earlier 1st I don't think a 3rd is going to move us up much. Maybe if we package our 1st with a really good prospect. A lot depends on how other teams tier a draft. If they have 2/3/4 guys they value pretty equally they're more likely to drop a couple spots. If they're down to looking at one or two guys then it is less likely and very unlikely that a 3rd rounder is going to move them enough. These can be very fluid situations--someone or two goes unexpectedly early and all of sudden players are falling and teams are reassessing player values and if you're up next that can mean making a hard decision on the fly. The Rangers IMO are in a good spot with their two firsts. If they really want a player that second first might move the first first up 5 or 6 spots--more than that I don't know.
I had to dig through a lot of old threads, but I found this which I typed up back in 2018:
Going back to 2005, these are the trades I could find of teams trading up solely using picks:
2017: 26th (DAL) for 29th + 70th (CHI)
2016: 26th (STL) for 28th + 87th (WSH)
2016: 18th + 3rd (WPG) for 22nd + 2nd (PHI)
2016: 16th + Datsyuk* (ARI) for 20th + 2nd (DET)
2016: 11th (OTT) for 12th + 3rd (NJD)
2015: 29th (CLB) for 33rd + 68th (TOR)
2015: 24th (PHI) for 29th + 61st (TOR)
2014: 20th + 179th (SJS) for 28th + 62nd (CHI)
2014: 28th (NYI) for 35th and 57th (TBL)
2013: 18th (SJS) for 20th + 58th (DET)
2012: 14th (BUF) for 21st + 42nd (CGY)
2011: 22nd (TOR) for 30th + 39th (ANA)
2011: 24th (OTT) for 35th + 48th (DET)
2010: 15th (LAK) for 19th + 59th (BOS)
2010: 22nd + 113th (MTL) for 27th + 57th (ARI)
2010: 30th (NYI) for 35th and 58th (CHI)
2009: 16th, 77th 182nd (MIN) for 12th (NYI)
2008: 5th (TOR) for 7th, 68th, 37th in 2009 (NYI)
2008: 7th (NSH) for 9th, 40th (NYI)
2007: 9th (SJS) for 13th, 44th, 87th in 2008 (STL)
2005: 8th (SJS) for 12th, 49th, 207th (ATL)
2005: 12th (NYR) fot 16th and 41st (ATL)
*Datsyuk already retired from the NHL, so this was just the cap hit traded to Arizona
And these are the teams who traded into the 1st (or up) with players involved:
2017: 7th overall + DeAngelo (NYR) for Raanta + Stepan (ARI)
2017: 31st overall + Sundqvist (STL) for Reaves + 2nd (PIT)
2017: 27th overall + cond. '18 1st + Lehtera (PHI) for B. Schenn (STL)
2016: 30th overall + '17 2nd (ANA) for Andersen (TOR)
2015: 21st overall (OTT) for Lehner + Legwand (BUF)
2015: 15th overall + two 2nd (BOS) for D. Hamilton (CGY)
2015: 13th overall + C. Miller + M. Jones (BOS) for Lucic (LAK)
2015: 16th overall + 2nd (NYI) for G. Reinhart (EDM)
2014: 24th overall + Sbisa, Bonino, 3rd (VAN) for Kesler + 3rd (ANA)
2014: 10th overall + Silfverberg + Noesen (ANA) for Ryan (OTT)
2013: 9th overall (VAN) for Schneider (NJD)
2013: 16th overall (BUF) + Hackett + J. Larsson for Pominville + 4th (MIN)