What comet's tail did Earth pass through before Halley's?
The quotation is probably referring to The Great Comet of 1861.
According to this Wikipedia article: https://en.wikipedia.org/wiki/C/1861_J1

The comet may have interacted with the Earth in an almost
unprecedented way. For two days, when the comet was at its closest,
the Earth was actually within the comet's tail, and streams of
cometary material converging towards the distant nucleus could be
seen.

This was reported in newspapers at that time:
https://www.newspapers.com/image/424095911/?terms=comet%20tail&amp;match=1
and also prior to the close approach of Borelli's Comet in 1903.
https://www.newspapers.com/image/85101704/?terms=%22%20borelli%20comet%22&amp;match=1

From  this archive of C/1861 J1 (Great Comet of 1861) from http://cometography.com/lcomets/1861j1.html


Further reading:

Great Comets in History by Donald K. Yeomans (April 2007) JPL/Caltech
Comet C/1861 J1 (Great comet) in JPL Horizons database
Periodic Comets Found in Historical Records
Do Comets C/1861 G1 (THATCHER) And C/1861 J1 (Great Comet)
Have a Common Origin?
International Comet Quarterly
Forward-Scattering Enhancement of Comet Brightness. I. Background and Model (large PDF, Int Comet Quart April 2007)
Forward-Scattering Enhancement of Comet Brightness. II. The Light Curve of C/2006 P1 (McNaught)  (large PDF, Int Comet Quart October 2007) also here
Forward-Scattering Enhancement of Comet Brightness. III. Prospects for C/2010 (Elenin) as viewed from the Earth and SOHO, STEREO-A, and STEREO-B spacecraft (large PDF, Int Comet Quart August 2011?) also here