Nov 22 (Reuters) - The Baltic Exchange"s dry bulk sea freight index climbed to its highest in a week on Monday, helped by improved rates for the larger capesize and panamax vessel segments. * The overall index (.BADI), which factors in rates for capesize, panamax and supramax vessels, gained 93 points, or 3.6%, to 2,645 points, rising for a third straight session. * The capesize index (.BACI) rose 274 points, or 7.6%, to its highest in three weeks at 3,884. * Average daily earnings for capesizes, which transport 150,000-tonne cargoes such as iron ore and coal, rose $2,272 to $32,210. * Steel futures prices in China jumped, with hot-rolled coils and construction rebar climbing more than 4% in intraday trade to narrow the gap with spot prices, as traders cheered a marginal improvement in consumption of industrial metals. * The panamax index (.BPNI) increased 9 points, or 0.4%, to 2,291, snapping an eight-session long losing streak. * Average daily earnings for panamaxes, which ferry 60,000-70,000 tonne coal or grain cargoes, increased by $85 to $20,620. * The supramax index (.BSIS) lost 2 points to six-month trough at 2,235. * Despite recent volatility in vessel rates, dry bulk spot rates are expected to stabilize in November-December before a "seasonally soft" first quarter of 2022, shipbroker Jefferies said in a weekly note. Reporting by Kavya Guduru in Bengaluru